iw^^^M^uVsi^.^W^WA.kr^i 


JVo. 

Division 
Range 

Shelf. 

Received 


mtoig 

OF   THE 


• 


EAST    EULES 


FOR    THE 


MEASUREMENT  OF  EARTHWORKS, 


BY  MEANS  OF  TIIE 


PRISMOIDAL    FORMULA. 


ILLUSTRATED     WITH     NUMEROUS     WOODCUTS,   PROBLEMS,    AND     EX- 
AMPLES,   AND    CONCLUDED   BY  AN    EXTENSIVE    TABLE 
FOR    FINDING    THE    SOLIDITY    IN   CUBIC 
YARDS  FROM  MEAN  AREAS. 

THE  WHOLH 

BEING  ADAPTED   FOR  CONVENIENT  USE  BY  ENGINEERS,  SURVEYORS. 

CONTRACTORS,    AND     OTHERS    NEEDING    CORRECT 

MEASUREMENTS  OF  EARTHWORK. 

L  I  B  R  A  II  V 


j  UNIVERSITY   OF 


CALIFORNIA. 

ELLWOOD    MORRIS,    CIVIL   ENGINEER. 


PHILADELPHIA: 

T.  R.  CALLENDER  &  CO.,  THIRD  AND  WALNUT  STS. 

LONDON:  TRUBNER  A  CO.,  60  PATERNOSTER  ROW. 
1872. 


Entered  according  to  Act  of  Congress,  in  the  year  1871,  by 

ELLWOOD  MORRIS,  CIVIL  ENGINEER,  OF  CAMDEN,  N.  J., 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington,  D.  C. 


RESPECTFULLY    DEDICATED 

TO    THE 

ENGINEERS,  SURVEYORS,  AND  CONTRACTORS 

OF 

THE  UNITED    STATES, 
BY  ONE  WHO  IS  WELL  ACQUAINTED 


THEIR  ABILITIES  AND  WORTH. 


TABLE  OF  CONTENTS, 

BY    CHAPTER,   AETICLE,    PAGE,  AND    REFERENCE    TO    ILLUSTRATIONS. 


CHAPTER  T. 


Art.    1. 


Art.    2. 
Art.    3. 


Art  4. 

Art.  5. 

Art.  6. 

Art.  7. 

Art.  8. 


Art.    9. 
Art.  10. 


Art.  11. 
Art.  12. 

Art.  13. 


PRELIMINAKY    PROBLEMS. 

PACE 

Of  the  Prismoid. — Origin  probably  ancient.  Prismoidal  Rule,  first  devised  and  7 
demonstrated  by  THOMAS  SIMPSON  (1750).  Generalized  and  improved  by  CHARLES 
HUITON  (1770).  Now  known  as  the  Prismoidal  Formula,.  A  Prismoid  denned, 
described,  and  illustrated  by  Fig.  I.  The  Prismoid,  a  frustum  of  a  wedge,  which 
latter  is  itself  a  truncated  triangular  Prism.  Example  of  rectangular  Prismoids, 
by  Simpson's  Rule  and  other  processes. 

Simpson's  I*rismoidal  RuU.  Applicable  chiefly  to  rectangular  and  triangular  10 
Prismoids,  with  examples  of  each.  Illustrated  by  Fig.  2.  11 

Huttoris  Prismoidal  Rules. — His  definition  of  a  Prismoid.  General  Rule,  in  which  12 
the  hypothetical  mid-section,  deduced  from  the  ends,  is  first  introduced  by  him, 
thus  generalizing  the  rule,  and  materially  extending  its  usefulness.  Particular 
Rule,  deduced  from  the  wedge.  Rectangular  Prismoid,  composed  of  two  wedges. 
Initial  Prismoids,  to  determine  generality  of  rule.  Applicability  of  Button's  Rule  13 
to  various  solids,  named  by  him.  Example  of  computation  by  General  and  Par-  15 
ticular  rules;  also,  by  Initial  Prismoids.  The  T  Prismoid,  example  of  its  compu-  16 
tation.  Figs.  3.  4,  and  5  illustrate  this  article. 

The  Prismoid  adapted  to  Earthwork.— By  Sir  John  Macneill,  C.  E.  (Fig.6),  his  figure      18 
of  a  prismoid  composed  of  a  prism  and  superposed  wedge.    Demonstrates  the 
Prismoidal  Rule,  and  uses  it  in  an  extensive  series  of  Tables  (1833). 

The  Prismoid  in  its  Simplest  Form. — Computed  by  several  rules.  Illustrated  by 
Fig.  7. 

Further  Illustration  of  MacneilUs  Prismoid. — Double  cross-sectioned  in  Fig.  8.  Con-      21 
cise  rule  for  earthwork  wedges,  confirmed  by  Chauvenet's  Geometry.    Brevity  of 
computation  of  earthwork  solids  on  level  ground,  by  combining  the  rules  for 
Wedge  and  Prism.    Wedges  applicable  also  to  irregular  ground. 

Trapezoidal  Prismoid  of  Earthwork,  considered  as  two  Wedges. — Button's  Particular  24 
Rule,  modified  for  earthwork.  Illustrated  by  Fig.  9,  and  by  examples. 

26 

Areas  of  Railroad  Cross-sections  (within  Diedral  Angles),  whether  Triangular,  26 
Quadrangular,  or  Irregular. — Triangles  and  Trapeziums  computable  by  their  28 
hights  and  widths,  taken  rectangularly,  however  they  may  be  placed  (see  Figs.  30 
10, 11, 12.  and  13).  Exact  method  of  equalizing  irregular  ground  surface,  by  a  32 
single  right  line  (see  Figs.  14  and  15).  Calculation  of  areas  of  crows-sections  33 
(Figs.  16  and  17).  Very  irregular  cross-sections  require  division  into  elementary  34 
figures,  these  to  be  separately  computed  and  totalized.  35 

Further  Illustration  of  the  Modification  of  Simpson's  Rule,  etc. — Example  illustrated 
by  Fig.  18.    Diagram  of  Simpson's  mid-section,  Fig.  19,  showing  its  curious  com-      37 
position.    Diagram  of  the  formulas  of  Simpson  and  Hutton,  Fig.  19%.    Formula 
deduced  from  it. 

Adaptation  of  the  Prismoidal  Formula  to  Quadrature,  Oubature,  etc.,  by  Simpson  40 
and  Hutton.— Explanation  of  it  by  Hutton,  under  the  head  of  Equidistant  Ordi-  42 
nates,  Fig.  20.  Example  of  two  stations  of  earthwork.  Figs.  21  to  26.  Coincidence  44 
with  the  Prismoidal  Formula.  Earthwork  calculations  by  Roots  and  Squares,  46 
explained  and  illustrated  by  examples,  and  by  Figs,  from  27  to  38 ;  also,  by  com-  47 
parison  with  other  methods.  Computation  of  the  passages  at  grade  points,  from  53 
excavation  to  embankment.  Figs.  39  to  42. 

Cross-sections  in  Diedral  Angles,  to  find  Mid-section,  etc. — Illustrated  by  Figs.  43  55 
and  44. 

To  find  Prismoidal  Mean  Area,  from  Arithmetical  or  Geometrical  Means,  from 
the  Mid-section,  or  by  Corrective  Fractions. — Examples,  illustrated  by  Figs.  45 
and  46. 

Applicability  of  the  Prismoidal  Formula  to  other  Solids. — The  three  round  bodies 
computable  by  it.  Examples  given  and  illustrated  by  Fig.  47.  This  formula  also 
applies  to  the  three  square,  or  angular  bodies. 


TABLE  OF  CONTENTS. 


Transformation  of  Solids  into  Equivalents  more  easily  Computable  Prismoidally.— 
Equivalency  explained.  With  examples  illustrated  by  Figs.  48,  49,  50,  and  51. 

Equivalence  of  some  important  Formulas,  with  that  for  the  I*rismoid. — Coincidence 
of  the  Pyramidal  and  Prisnioidal  Formulas  to  a  certain  extent,  and  similitude  of 
their  results  within  certain  limits.  Illustrated  by  Fig.  52  (in  projection),  and 
followed  by  examples  calculated  both  Prismoidally  and  Pyramidally,  and  by  other 
rules;  all  equivalent  in  solidity  upon  level  ground. 

Summary  of  Rules  and  Formulas  in  Chapter  I. — Indicated  by  numbers  from  I.  to 
XII.,  and  by  reference  to  the  various  articles  of  the  chapter  in  which  they  may 
be  found. 

CHAPTER.  II. 

FIRST  METHOD  OF  COMPUTATION,  BY   MID-SECTIONS  DRAWN  AND  CALCULATED  FOB  AREA 
ON  THE   BASIS   OF  HUTTON'8  GENERAL   RULE. 

encrality  and  accuracy  of  the  Prismoidal  Formula.  Manner  of  collecting  the  field 
data.  Kules  for  cross-sectioning.  Correction  for  curvature.  Vital  importance  of 
judicious  cross-sectioning.  Ground  to  be  sectioned  so  as  to  reduce  it  practically 
to  plane  surfaces.  Warped  surfaces  and  opposite  slopes  deemed  inadmissible  in 
general.  Expression  for  the  Prismoidal  Formula,  as  generalized  by  HUTTON. 
Classification  of  the  ground  surface.  Mensuration  by  cross-sections,  long  used  by 
mathematicians,  and  early  adopted  by  engineers. 

Examples  in  computation  by  our  First  Method.    Illustrated  by  Figs.  53  to  64. 

Connected  calculation  of  contiguous  portions  of  Excavation  and  Embankment,  with 
the  passage  from  one  to  the  other.  Illustrated  by  Figs.  65  to  71.  End  sections  to 
have,  or  receive,  the  same  number  of  sides  before  calculation.  Observations  on 
mid  sections.  Importance  of  verifying  all  calculations.  The  Prisnioidal  Formula, 
the  standard  test  for  solidity.  Correction  for  centres  of  gravity  referred  to  aa  a 
refinement  promoting  accuracy,  but  not  employed  by  engineers.  This  First 
Method  easily  applicable  to  masonry  calculations.  Borden's  Problem,  a  striking 
example  of  bud  cross-sectioning. 


CHAPTER  III. 

SECOND  METHOD   OF  COMPUTATION,   BY  HIOHT8  AND  WIDTHS,  AFTER  SIMPSON'S 
ORIGINAL    RULE. 

Prismoidal  Rule,  originated  and  demonstrated  by  Simpson  (1750)  for  rectangular 
prismoids.  Its  transformation  for  triangular  areas.  Modification  by  direct  and 
cross  multiplication. 

Examples  of  computation  by  our  Second  Method.    Illustrated  by  Figs.  72  to  75. 

Observations  upon  Simpson's  Rule  Ground  surfaces  to  be  equalized  to  a  single 
line.  Peculiar  solid  where  Simpson's  Rule  fails.  See  Figs.  81  and  82.  Examples 
of  rough  ground  equalized  by  a  single  line.  Illustrated  by  Figs.  43  and  44. 
Example  of  a  heavy  embankment,  from  Warner's  Earthwork,  on  uniform  ground, 
sloping  15°  transversely. 

CHAPTER  IV. 

THIRD  METHOD  OF  COMPUTATION,  BY  MEANS  OF  ROOTS  AND  SQUARES  J  A  PECULIAR  MODI- 
FICATION OF  THE  PRISMOIDAL  FORMULA,  WHICH  WILL  BE  FOUND  IN  PRACTICE  TO  U 
BOTH  EXPEDITIOUS  AND  CORRECT,  IN  ORDINARY  CASES. 

Enunciation  of  thfi  formnla  used.  Comparison  with  the  Prismoidal  Formula.  Use 
of  Simpson's  rule  for  cubatnre.  as  adopted  by  Hutton.  Skeleton  tabular  arrange- 
ment of  data.  Tabulations  for  solidity.  Numbered  places  for  end-sections  and 
mid-sections.  Example  of  a  heavy  embankment  and  rock-cut  (profiled  in  Fig. 
76),  and  each  computed  in  a  body  by  this  method.  Relations  of  the  squares  of 
lines,  or  parts  of  lines.  Re-computation  by  Roots  and  Squares  of  the  examples  of 
Chapter  II.  (illustrated  by  Figs.  53  to  64),  showing  their  close  agreement  with 
this  method. 

CHAPTER   V. 

FOURTH  METHOD  OF  COMPUTATION,  BY  REGARDING  THE  PRISMOID  AS  BEING  COMPOSED 
OF  A  PRISM,  WITH  A  WEDGE  SUPERPOSED,  OB  OF  A  WEDGE  AND  PYRAMID  COMBINED. 

Macneill's  wedge  superposed  upon  a  prism  to  form  a  prismoid.    Formula  for  wedge 

and  prism  to  find  solidity  of  prismoid.    Discussion  of  the  inclined  wedge  (Fig.  79), 

and  calculation  of-it. 

Rules  for  computation  by  Wedge  and  Priam. 
Discussion  of  the  wedge  in  connection  with  Chauvenet's  Theorem.    Illustrated  by 

9  small  ciits.    Showing  the  mode  of  computing  various  wedges. 
Examples  of  Wedge  and  Prism  computation.    Illustrated  by  Figs.  80, 14,  43,  and  44. 
Peculiar  solid  cross-sectioned  (Fig.  81),  and  in  projection  (Fig.  82).    Examples  of 

modes  of  computing  it. 


TABLE  OF    CONTENTS. 


Art.  30. 


Aw.  31. 

Art.  32. 

Ait.  33. 
Art.  34. 


Art.  35. 

Art.  36. 

Art.  37. 
Art.  38. 


?he  Rhomboidal  Wedge  and  Pyramid  introduced.  Examples  of  calculation  by  using 
them  to  find  the  solidity  of  a  prismoid. 

e  method  of  using  the  Rhomboidal  Wedge  and  Pyramid  combined.    (Figs.  81 
and  82.) 

Any  irregular  earthwork,  within  certain  limits,  computable  by  Wedge  and  Pyramid. 
Process  of  computation.  Notation  of  rule.  The  Rule  itself.  Its  limits  of  range. 

Examples  of  computation  by  Wedge  and  Pyramid ;  direct  and  reverse.  Re-com- 
puting by  this  method  the  examples  of  Chapter  II.,  and  showing  their  near  coin- 
cidence. Illustrated  by  Figs,  from  53  to  64,  in  Chapter  II. 

Ixample  of  heavy  embankment  (from  Warner's  Earthwork),  computed  by  Wedge 
and  Pyramid. 

eculiar  solid  again ;  illustrating  Case  2,  of  the  rule  for  Wedge  and  Pyramid,  which 
correctly  gives  its  volume. 

CHAPTER  VI. 

PROFESSOE  GILLESPIE'S  FOUR  USUAL  RULES,  WITH  THEIR  CORRECTIONS,  AND  A  COMPARI- 
SON OF  HIS  CHIEF  EXAMPLE  WITH  OUR  THIRD  METHOD  OF  COMPUTATION — BY  ROOTS 
AND  SQUARES. 

Che  late  Professor  Gillespie's  ability  and  labors.  The  four  usual  rules,  which  he 
found  in  use.  1.  Arithmetical  Average,  a  rule  very  much  used,  but  very  incor- 
rect. The  correction  necessary  for  true  results.  Tabulation  of  an  example  by  the 
Arithmetical  Average,  with  Gillespie's  correction.  2.  Middle  Areas,  rule  explained 
and  formula  for  correction  to  give  true  solidity.  Tabulation  of  example  by  Middle 
Areas,  with  correction.  3.  The  Prismoidal  Formula,  as  generalized  by  Hutton. 
The  test  rule  for  all.  Tabulation  of  example  by  it.  4.  Mean  Proportionals,  or 
Geometrical  Average.  Explanation  of  the  rule.  Gillespie's  condemnation  of  Geo- 
metrical Average,  and  of  Equivalent  Level  Eights,  evidently  hasty. 

Ele-computation  of  example  by  Geometrical  Average,  using  the  grade  triangle,  and 
tabulating  to  intersection  of  side-slopes.  Solidity  the  same  as  given  by  the  Pris- 
moidal Formula,  as  calculated  by  Gillespie  himself.  Generalization  from  which 
this  rule  flows.  Equivalent  Level  Hights.  Tabulation  of  an  example  by  them  to 
intersection  of  side-slopes.  Resulting  in  solidity  the  same  as  by  the  Prismoidal 
Formula,  calculated  by  Professor  Gillespie. 

Five  reliable  rules  for  earthworks,  deduced  from  this  Chapter.  Brief  reference  to 
the  Core  and  Slope  Method,  one  of  the  earliest  proposed,  and  often  re-produced, 
but  not  in  general  use. 

Tabulated  comparison  of  Gillespie's  example  by  the  method  of  Roots  and  Squares, 
agreeing  with  the  Prismoidal  Formula,  and  may  be  used  with  Simpson's  Multi- 
pliers. Gillespie's  examples  usually  refer  to  level  ground,  and  hence  are  easily 
computed. 

.CHAPTER  VII. 

PRELIMINARY  OR  HASTY  ESTIMATES,  COMPUTED  BY  SIMPSON'S  RULE  FOR  CUBATURE. 

Nature  of  Preliminary  Estimates.  Rough  and  expeditious,  but  giving  quantities 
proximately  correct.  Quantities  to  be  always  full,  but  never  excessive.  Simp- 
son's rule  for  cubature.  Explanation  and  notation.  Mode  of  numbering  the 
cross-sections,  even  and  odd.  Cuts  and  Banks  to  be  computed  separately,  each 
in  a  body.  General  Mean  Area  to  be  found,  and  multiplied  by  length  for  volume. 

Simpson's  Multipliers,  their  convenience  and  application.  Diagrams  indicating  a 
probable  construction  of  the  rule  for  cubature,  and  its  intimate  connection  with 
the  Prismoidal  Formula  (Figs.  77  and  78.) 

Rough  profile  to  be  sketched,  involving- bights  and  ground  slopes.  Computation  of 
a  Bank  by  Simpson's  Multipliers.  Rock-cut  calculated  in  the  same  way.  Both 
illustrated  by  Fig.  76. 

Reference  to  the  grade  prism.  Earth  cutting,  Fig.  29,  computed  by  both  Roots  and 
Squares  and  the  Hasty  Process,  differing  only  one  and  a  half  per  cent. 

Tables  (four  in  number)  for  use  in  Preliminary  or  Hasty  Estimates. 


PAGH 

133 


Following  this  Chapter,  and  closing  the  Book,  will  be  found  an  extensive  TABLE  OF 
CUBIC  YARDS  to  mean  areas  for  100  feet  stations  (entirely  clear  of  error,  it  is 
believed),  giving  the  Cubic  Yards  for  every  foot  and  tenth  of  mean  area  from  0  to 
1000,  by  direct  inspection.  And  being  computed  accurately  to  three  decimal 
places,  ranges  correctly  up  to  100,000  square  feet  of  mean  area,  or  to  a  cut  1000 
feet  wide,  and  100  feet  deep.  Table  preceded  by  explanations,  and  some  examples 
of  its  use.  This  Table  also  operates  as  a  general  one  for  the  conversion  of  any  sum 
of  cubic  feet  into  Cubic  Yards,  by  simply  dividing  by  100  and  using  the  quotient 
as  a  mean  area,. 


UNIVERSITY  OF 

CALIFORNIA. 


EASY    RULES 

FOR  THE 

MEASUREMENT    OF    EARTHWORKS, 

BY  MEANS  OF  THE  PRISMOIDAL  FORMULA. 


CHAPTER  L 

PRELIMINARY  PROBLEMS. 

1.  Of  the  Prismoid. — Although  this  solid  probably  originated  \vith 
the  ancient  geometers — THOMAS  SIMPSON  (1750),  an  eminent  mathe- 
matician of  the  last  century,  appears  to  have  been  the  first,  in  later 
days,  to  demonstrate  the  rule  for  its  solidity,*  now  accepted  by 
modern  mensurators ;  and  he  was  soon  followed  by  Hutton,  in  his 
quarto  treatise  on  Mensuration,f  who  by  another  process  again 
demonstrated  the  Prismoidal  Rule,  and  at  the  same  time  laid  the 
foundations  of  modern  mensuration,  in  a  manner  so  solid,  that  it  has 
come  down  to  our  time,  through  various  editors  and  commentators, 
substantially  (in  many  cases  literally)  the  same  as  established  by  Hut- 
ton  in  his  famous  work  of  1770. 

Simpson's  rule  for  the  prismoid  has  been  variously  transformed, 
and  written,  and  is  now  generally  known  by  the  name  of  the  prismoi- 
dal  formula,  of  which  we  will  give  hereafter  the  usual  expressions,  as 
well  as  some  useful  modifications,  the  same  in  substance,  but  often 
more  convenient  for  practical  purposes. 

The  solid  called  a  Prismoid  (from  its  general  resemblance  to  a 
prism,  and  in  like  manner  named  from  its  base,  triangular,  rectangu- 
lar, trapezoidal,  etc.)  is  a  body  contained  between  two  parallel  planest 

*  Simpson's  Doctrine  of  Fluxions.     (1750),  8vo,  London. 
f  Button's  Mensuration.    (1770),  4to,  Newcastle  upon  Tyne. 

7 


8 


MEASUREMENT  OF  EARTHWORKS. 


its  hight  being  their  perpendicular  distance  apart,  its  ends  rectangles* 
and  its  faces  plane  trapezoids ; — and  this  seems  to  be  a  sufficient  defini- 
tion. As  to  such  form,  all  prismoids  may  be  reduced  or  made  equiva- 
lent; but  although  this  simple  definition  answers  our  purpose  of  intro- 
ducing the  rectangular  prismoid,  HUTTON'S,  Art.  3,  is  the  authorita- 
tive one. 

This  solid  is  usually  the  frustum  of  a  wedge ;  but  as  the  proportions 
of  the  ends  are  changed,  it  may  become  a  frustum  of  a  pyramid,  a 
complete  pyramid,  a  wedge,  or  a  prism  ;  and  hence  it  is  indispensably 
necessary  that  the  rule  for  its  solidity  should  also  hold  for  all  these 
solids,  which,  in  fact,  it  does. 

The  ends  may  be,  and  often  are,  irregular  polygons,  but  they  must 
always  coincide  with  the  limiting  parallel  planes ;  and  though  the 
solid  may  be  quite  oblique,  its  hight  must  be  taken  normal  to  the 
end  planes.  The  faces  are  usually  straight  longitudinally,  but  this 
condition  is  not  absolute,  since  the  remarkable  formula,  deduced  from 
the  prismoid  for  its  solidity,  applies  as  well  to  the  volume  of  many 
curved  solids  in  an  extraordinary  manner,  of  which  the  limits  are  not 
yet  known,  though  more  than  a  century  has  elapsed  since  Simpson 
developed  it.  _____ 

The  mid-section,  inclu- 
ded by  the  usual  prismoi- 
dal  formula,  must  be  in 
a  plane  parallel  to,  and 
equally  distant  from,  those 
containing  the  ends,  and 
is  deduced  from  the  arith- 
metical average  of  like 
parts  in  them.  It  is  en- 
tirely hypothetical,  or  as- 
sumed for  the  purposes  of 
computation,  and  has  no 
actual  existence  in  the 
body  itself. 

The  rectangular  pris- 
moid (usually  regarded  as 
the  elementary  figure  of 
this  solid)  is  a  frustum 
of  the  wedge. 

(a.) Thus  the  prismoid  AB  (Fig.  1)  is  a  frustum  of  the 

wedge  AEC. 


CHAP.   I.— PRELIM.   PROBS.-ART.  1.  9 

The  wedge  AEG  itself  being  a  triangular  prism,  truncated  twice, 
the  rectangular  prismoid  then  is  a  triangular  prism,  trebly  truncated : 
1st,  by  two  cutting  planes,  reduced  to  a  wedge;  and  2nd,  by  another 
plane,  to  a  prismoid  (AB),  the  latter  being  parallel  to  the  base,  and 
by  its  section  forming  the  top  of  the  solid  at  B. 

The  prismoid,  therefore,  may  be  computed  as  a  truncated  triangu- 
lar prism  or  wedge,  and  the  part  cut  off  deducted,  in  like  manner  as 
the  frustum  of  a  pyramid  may  be  calculated  as  though  the  pyramid 
was  complete,  and  then  the  truncated  part  computed  separately  and 
subtracted,  leaving  only  the  solidity  of  the  frustum,  subject,  like  the 
prismoid,  to  calculation,  by  more  concise  rules,  if  expedient. 

Referring  now  to  Fig.  1. 

Let  Abode/be  the  original  triangular  prism,  truncated  right  and 
left  by  planes  passing  through  A  b  and  ef,  reducing  it  first  to  the 
wedge  AE ;  and  secondly,  by  passing  the  plane  B  2,  parallel  to  the 
base  eb,  leaving  as  the  residual  solid,  after  three  truncations,  the 
Prismoid  AB. 

Then,  in  the  wedge  AEC,  the  right  section  has  a  base  of  4,  a  hight 
of  12,  and  area  of  24,  which,  multiplied  by  £  the  sum  of  the  lateral 
edges  *  (or  6§),  gives  a  solidity  of  160 ;  while  the  wedge  BCE,  cut 
off,  has  a  base  of  2,  and  hjght  of  6,  in  its  right  section,  or  area  of  6, 
which,  multiplied  by  i  the  sum  of  its  lateral  edges  (or  5i),  gives  a 
volume  of  32. 

Now,  160  —  32  =  128,  the  solidity  of  the  Prismoid  AB,  as  is  shown 
(more  concisely)  05  follows : 

By  Simpson's  Rule — 

lit*.     Widths. 

Base, 8  X  4  =     32 

Top, 6  X  2  =     12 


ums,  equivalent  to  )    +  ,   ^  R          ft  . 

-J  f     -I*     X     O    —        O-4 

id.  sec.,    .     .     .     .  J 


Product  of  sums,  equivalent  to 

A    j.'  -J 

4  times  mid. 

128 

Multiplied  by  i  h.     .    .i.T   ;.**..=       1 

Solidity,  .    .    .  :.  >/}  .  .'J-  .  *;»*«;.    .  =  128 

(The  same  as  above.) 

Precisely  the  same  result  is  also  reached  by  means  of  the  centre  of 
gravity  of  the  right  section,  flowing  with  that  section  along  a  line 

*  Chauvenet's  Geom.  (1871),  vii.  22.  A  wedge,  whether  trapezoidal  or  rectangular, 
being  merely  a  truncated  triangular  prism,  this  rule  of  Chauvenet's  is  probably  the 
most  concise,,  and  fast  for  ordinary  uw. 


10 


MEASUREMENT  OF  EARTHWORKS. 


curved  with  an  infinite  radius,  according  to  Button's  Problem.*  The 
right  section  of  the  prismoid  AB  (Fig.  1)  is  a  plane  trapezoid  (18  in 
area),  of  which  (from  the  dimensions  given  in  the  figure)  the  centre 
of  gravity  is  found  in  a  perpendicular  line,  drawn  from  the  middle 
of  A  6,  and  at  the  distance  of  21  feet  vertically  from  it.  Now,  the 
length  of  a  straight  line,  drawn  from  face  to  face  of  the  prismoid, 
parallel  to  the  plane  of  the  base — also  to  its  edges — and  at  a  vertical 
distance  of  2f  feet,  will  be  7J  feet,  by  which  the  right  section  (18) 
being  multiplied,  we  have  for  the  solidity  =  128,  as  before. 

2.  THOMAS  SIMPSON'S  Prismoidal  Rule. — In  his  work  on  Fluxions 
and  their  Applications 
(1750),  Simpson  demon- 
strates the  following  rule 
for  the  solidity  of  a  pris- 
moid, referring  to  Fig.  2. 

This  rule  for  the  pris- 
moid, as  demonstrated 
by  Simpson,  renders  the 
formation  of  the  hypo- 
thetical mid-section  un- 
necessary, though  con- 
taining it,  in  effect,  as 
marked  upon  the  figure, 
for  illustration. 

wA. 


Simpson's  Rule  is  as 
follows: — Fig.  2. 
(AB  X  AD)  +  (EH  X  EF)  +  (AB+EH  X  AD  +  EF)  X 

i  h  =  Solidity, (I.) 

Or, 

/hight  X  width \     ,     /hight  X  width  \     , 
\     of  one  end,     j  '      (     -*-Al- J 


of  one  end,     / 

/  sum  of  bights  X   sum  of  widths  \ 
\  of  both  ends,  / 


of  other  end,    ) 


/T  \ 

(I.) 


Here  AB  X  AD  =  area  of  base.  EH  X  EF  =  area  of  top.  While 
the  product  of  their  sums  =  (AB  +  EH)  X  (AD  +  EF)  =  four 
times  the  area  of  the  mid-section. 


*  Button's  Mens.  (1770),  part  iv.  sec.  3. 


CHAP.  I.— PRELIM.  PROBS.— ART.  2.  11 

EXAMPLE  1. 

Let  AB  and  EH  be  called  the  widths,  AD  and  EF  the  highte, 
and  take  the  dimensions  marked  upon  Fig.  2.  Then,  by  Simpson's 
rule,  we  have  for  the  solidity  of  this  rectangular  prismoid  the  fol- 
lowing : 

Widths.       Ht8. 

20  X  16  =     320  =  area  of  base. 
18  X  12  =     216  =      do.     top. 

Sums  of  hts.  and  widths  =  38  X  28  =  1064  =  four  times  mid-sec. 

1600  =  sum  of  areas. 
Multiplied  by  i  h  =  2e4,     .     .     .     .  =         4  =  i  h. 

Solidity, =  6400  =  volume. 

(a.) The  above  is  a  rectangular  prismoid,  or  one  in  which  all  the 

parallel  sections  are  rectangles.  Now,  suppose  this  prismoid  to  be 
cut  diagonally  by  a  plane,  FHBD,  dividing  it  into  two  triangular 
prismoids,  each  equal  to  the  other,  and  to  one-half  of  the  rectangular 
prismoid. 

Then  (AB  X  AD)  =  double  the  base;  (EH  X  EF)  =. double 
the  top;  and  (AB  -f  EH)  X  (AD  +  EF)  =  eight  times  the  mid- 
section. 

Hence,  Simpson's  rule,  thojugh  applicable  to  any  prismoid,  by 
reducing  the  ends  to  equivalent  rectangles,  seems  especially  suitable  to 
triangular  prismoids,  since  the  double  area  of  every  triangle  is  equal 
to  the  product  of  its  bight  and  width,  taken  rectangularly;  while 
the  product  of  the  sums  of  those  bights  and  widths,  multiplied  to- 
gether, gives  eight  times  the  area  of  the  mid-section,  without  the  ne- 
cessity of  forming  it  by  arithmetical  averages. 

Accordingly,  with  triangular  sections,  a  slight  transformation  of 
this  rule  will  often  be  more  convenient  for  use  with  given  areas. 

Thus, 

Let  double  the  area  of  the  base =    2  b. 

top   .    .-'.;   v   .    .     .  =    2  t. 

Eight  times  the  area  of  the  mid-sec,    v  ^V   *;  n'   .  =    8   m. 
And  the  final  divisor  (12),  or  if  used  as  above,     .  =  ^  h. 

Then,  to  find,  in  the  first  instance,  the  mean  area  of  the  prismoid. 

We  have  the  formula,  -        — — —        -  =  mean  area  .     .     (II.) 

And  this  mean  area,  being  multiplied  by  the  bight  or  length  (h), 
of  the  whole  prismoid  between  the  end  planes,  gives  the  solidity. 


12  MEASUREMENT  OF  EARTHWORKS. 

Thus,  in  the  case  of  the  two  triangular  prismoids,  into  which  the 
diagonal  plane  FB  (Fig.  2)  divides  Simpson's  rectangular  prismoid, 
we  have,  by  taking  the  dimensions  marked  upon  the  figure, — the  fol- 
lowing : 

EXAMPLE  2. 

Calculation  of  the  triangular  prismoid  ABDFHE,  or  of  its  equal 
GD  =  3200,  Solidity. 

Hts.        Widths. 

16  X  20  =     320     =  2  b. 

12  X  IS  =     216     =  2  t. 


Sums,  .    .  28  X  38  =  1064     =  8  m. 

12)1600 
Mean  area,     .     .  =     1331  X  h  =  24  =  3200,  Solidity. 

And  3200  X  2  =  6400  =  the  solidity  of  the  whole  rectangular 
prismoid,  as  above. 

3.  CHARLES  HUTTON'S  Prismoidal  Rules. — In  his  famous  quarto 
Mensuration  (Newcastle-upon-Tyne,  1770),  Hutton  gives  the  follow- 
ing definition : 

"A  prismoid  is  a  solid  having  for  its  two  ends  any  dissimilar  par- 
allel plane  figures  of  the  same  number  of  sides,  and  all  the  sides  of 
the  solid,  plane  figures  also." 

He  adds :  "  It  is  evident  that  the  sides  of  this  solid  are  all  trape- 
zoids ;"  and  :  "  If  the  ends  of  the  prismoid  be  bounded  by  curves,  as 
ellipses,  etc.,  the  number  of  its  sides,  or  trapezoids,  will  be  infinite, 
and  it  is  then  called,  sometimes,  a  cylindroid." 

Hutton  gives  two  rules  for  the  solidity  of  the  body  (so  defined), 
one  general,  and  the  other  he  calls  the  particular  rule — he  also  indi- 
cates a  third,  by  means  of  initial  prismoids,  which,  by  a  little  develop- 
ment, can  be  made  quite  useful. 

Button's  General  Mule. 

"  To  the  sum  of  the  areas  of  the  two  ends  add  four  times  the  area 
of  a  section  parallel  to,  and  equally  distant  from,  both  ends,  mul- 
tiply the  last  sum  by  the  hight,  and  i  of  the  product  will  be  the 
solidity, (III.) 

In  this  shape,  and  nearly  in  the  same  words,  through  Bonnycastle, 
and  other  writers  on  Mensuration,  the  Prismoidal  Formula  has  come 
down  to  our  time. 

In  the  work  above  cited,  Hutton  also  (part  iv.  prop.  3)  shows  that 


CHAP.  I.— PRELIM.   PROBS.-ART.  3. 


13 


Fig.  3 


t  of  the  sum  of  the  end  areas,  and  four  times  the  mid-section,  gives 
the  mean  area  of  any  prismoidal  solid,  which,  multiplied  by  its  length, 
will  equal  the  solidity. 

The  particular  rule,  referred  to  above,  is  directly  deduced  from  that 
given  by  him  for  the  solidity  of  a  wedge. 

Thus,  referring  to  Fig.  3  (copied  by  us  from  the  original  work 
of  1770). 

Hutton  says,  where  L  and  I  represent  two  corresponding  dimen- 
sions of  the  end  rectangles,  B  and  b  the  others,  and  h.  the  bight  or 
length  of  the  prismoid, 

Then,  

(217+1  X  B  +  27TL  X  b)  X  t  h  •=  Solidity, 

— which  is  the  particular  rule,    .     .•    .     .     .     .-    .     (IV.) 

A  note,  on  page  163,  referring  to  this, 
says: 

"It  is  evident  that  the  rectangular 
prismoid  is  composed  of  two  wedges, 
whose  bases  are  the  two  ends  of  the 
prismoid,  and  whose  hights  are  each  equal 
to  that  of  the  prismoid." 

It  might  be  added,  that  the  edges  of 
these  two  wedges  are  formed  by  two 
diagonally  opposite  sides  of  the  rectangu- 
lar ends. 

Hutton  notes  also, 

That  -      -  =  M,  and  -      -  =  m,  the  sides  of  the  mid-section,  so 

_  L 

that  the  correspondence  of  the  General  and  Particular  Kules  becomes 
evident. 

(a.) At  page  164  of  the  quarto  Mensuration,  cited  above, 

reference  is  made  to  the  General  Rule  as  follows : 

"  This  rule  will  serve  for  any  prismoid  or  cylindroid,  of  whatever 
figure  the  ends  may  be,  inasmuch  as  they  may  be  conceived  to  be  com- 
posed of  an  infinite  number  of  rectangular  prismoids.  Which  is  the 
General  Rule." 

This  method  of  considering  any  prismoid  to  be  composed  of  a  great 
number  of  rectangular  prismoids,  of  the  same  common  length,  has  pre- 
vailed from  Hutton's  time  down  to  the  present  day. 

Thus,  we  find  in  Davies  Legendre,*  chapter  on  the  Mensuration 


*  Davies  Legendre.     (1853),  8vo  :  New  York. 


14 


MEASUREMENT  OF  EARTHWORKS. 


of  Solids,  in  treating  of  prismoids,  where  he  copies  Hutton's  figure, 
and  both  Particular  and  General  Rules, — the  following  : 

"  This  rule  (the  general  one}  may  be  applied  to  any  prismoid  what- 
ever. For  whatever  the  form  of  the  bases,  there  may  be  inscribed  in 
each  the  same  number  of  rectangles,  and  the  number  of  these 
rectangles  may  be  made  so  great  that  their  sum  in  each  base  will 
differ  from  that  base  by  less  than  any  assignable  quantity.  Now,  if 
on  these  .rectangles  rectangular  prismoids  be  constructed,  their  sum 
will  differ  from  the  given  prismoid  by  less  than  any  assignable  quan- 
tity. Hence,  the  rule  is  general." 

In  his  remarkable  chapter  on  the  cubature  of  curves  (Mens.,  part 
iv.  page  457),  Hutton  shows  that  the  prismoidal  formula  is  applica- 
ble to  the  frusta  of  all  solids 
generated  by  the  revolution 
of  a  conic  section  (as  well 
as  to  the  complete  solids); 
also,  to  all  pyramids  and 
cones,  and  in  short  to  all 
solids  (right  or  oblique),  of 
which  the  parallel  sections 
are  similar  figures. 

We  will  now  illustrate 
Hutton's  Rules,  by  means 
of  a  figure  and  examples,  to 
find  the  solidity  of  a  pris- 
moid, with  very  dissimilar 
(See  Fig.  4.)  * 


Kg.  4. 


1.  By  General  Rule* 

40  X  30  =  1200  =  b. 
80  X    4  »    320  =  t. 
60  X  17  X    4  =  4080  =  4  m. 


6)5600 


Multiplied  by  ll 
Solidity 


60 


56000 


2.  By  Particular  Rule. 

As -two  Wedges. 

40  80 

2  2 

160 
40 

200 
4 

800 
10 


48000 
8000 


8000 


Solidity  —  56000  of  whole  pris- 
moid. 


CHAP.   I.— PRELIM.    PROBS.— ART.   3. 


15 


3.  By  means  of  Initial  Prismoids  ......  (V.)  (To  be  further  explained.) 

(1)  Areas  of  ends,  b  =  1200,  and  t  =  320. 
,ON    f  Rights  =  30  )  ,    =    4  K 


(3)  Assumed  squares  in  larger  end,  1200  of  1  X 

t         320 

(4)  Ratio  of  ends,  -  =  —  =-2667. 


UNIT  E  R  S  I  T 


80 


(5)  Proportional  rectangles  in  small  end  (1200  in  number),  — 

40 


2, 


•—  =  -13333,  2  X  '13333 

oO 


'26667  =  area  of  these,  being  equiva- 
lent to  the  ratio  of  the  ends  1  to  '2667.     [See  (4).] 


1-1-2 

(6)  Mid-section,  dimensions  of  proportional  rectangle,  --  =  1 


1  +  -13333 


•5667,  and  1  '5  X  '5667  =  '85  =  rectangular  area  of 


b'  =1  X  1 


mid-section  of  initial  prismoid. 

Then  for  the  solidity  of  the  initial  prismoid,  by  General 
Rule. 

Call  these  areas 
b',  m',  and  t',  to 
distinguish  them 

4m' =   -85  X  4    .  =    3-4 

f    =   -13333  X  2  =      -26667 
6)  4-66667 

Mean  area,    .     .     .    .  =      -77778 
Multiplied  by  h    .     .  =  60 

Volume  of  one, *=    46-66680 

Mult,  by  No.  initial  prismoids,  assumed  =  1200 


(7) 


from  those  of  the 
main  solid. 


Solidity  of  the  whole  prismoid,  as  above  =  56000-16000 
In  computing  initial  prismoids  it  is  necessary  to  em- 
ploy sufficient  decimals,  but  4  or  5  places  are  usually 
enough. 


(b.) These  initial  prismoids  are  supposed  to  be  constructed  upon 

small  rectangles  in  the  two  ends,  equal  in  number  in  each,  and  of  pro- 
portional areas. 

In  the  base,  or  larger  end  (though  either  end  may  be  used),  it  will 
be  most  convenient  to  assume  these  to  be  squares  formed  upon  the 
unit  of  measure,  while  at  the  top  they  must  be  rectangles  proportional 
both  in  dimensions  and  area,  by  the  view  we  have  herein  taken  (as 
indicated  at  (5)  above). 


16 


MEASUREMENT  OF  EARTHWORKS. 


The  end  areas  of  the  main  prismoid  being  always  given,  or  com- 
putable, they  must  be  proximately  reduced  to  rectangles  before  we 
can  properly  apply  the  principle  of  initial  prismoids  to  calculate,  or 
verify,  their  solidity ; — and  the  solid  will  then  become,  in  effect,  a 
rectangular  prismoid  like  those  of  Simpson  and  Hutton. 

In  doing  this,  it  will  be  sufficient  to  dermine  a  width  and  hight, 
apparently  proportional  to  the  shape  of  the  cross  section  (which  in 
some  species  of  earthwork  is  extremely  irregular), — but  this  hight 
and  width  must  be  such  that,  used  as  factors,  they  reproduce  the 
given  area,  even  though  of  themselves  they  may  not  be  exactly  geo- 
metrical equivalents,  for  the  dimensions  of  the  section. 

Having  thus  (as  it  were)  rectified  the  solid  proximately,  we  may 
proceed  with  it  as  a  rectangular  prismoid,  by  the  method  of  initial 
prismoids,  briefly  as  follows : — Determine  the  rectangular  hights  and 
widths,  such  as  will  proximate  the  figure,  and  by  multiplication  reproduce 
the  areas.  Assume  one  end  as  base,  to  be  divided  into  squares  of  super- 
ficial units,  and  the  others  into  proportional  rectangles;  upon  these  con- 
struct (or  imagine)  ini- 
tial prismoids,  and  having 
ascertained  the  volume 
of  one,  multiply  by  num- 
ber, for  solidity  of  main 
prismoid,  as  shown  in  de- 
tail above.  .  .  .  (V.) 

(C.) We  will 

further  illustrate  tiiis 
subject  by  presenting  an 
outline  of  a  T-shaped 
prismoid ;  a  solid  (Fig. 
5),  with  a  figure  so  pecu- 
liar that  none  of  the 
usual  methods  of  averag- 
ing could  even  proximate 

its  solidity, which 

can  only  be  dealt  with   by  the  Prismoidal  Formula,  or  some  cog- 
nate rules. 

This  we  will  calculate  as  a  prismoid  by  Simpson's  General  Rule, 
by  Hutton's  Particular  Rule,  and  by  the  Method  of  Initial  Prismoids. 


nu&seo 


CHAP.   I.— PRELIM.    PROBS.— ART.    3. 


17 


By  Hutton's  Particular  Rule. 


100 
2 

200 


208 
6 

1248 
100 


As  two  Wedges. 


8 

2 

16 

100 

116 
50 

5800 
100 


6)  124800   6  )  580000 

20800      96666J 
20800 

Solidity  =  117466S 


By  Simpson's  General  Rule. 

As  a  Rectangular  Prismoid. 
Hts.        Wds. 
6   X   100  .    .   = 

50  X      8  .  .  = 


Sums,  56  X  108  = 
4  times  mid-sec. 


600 
400 

6048 
7048 


Solidity,.  .  .  =  1174661 


By  the  Method  of  Initial  Prismoids. — Let  their  number  be  400,  the 
same  as  the  superficies  of  A.  Suppose  them  constructed  upon 
squares  at  A.  (on  a  side  equal  to  the  unit  of  measure),  and  upon  pro- 
portional rectangles  at  BC. 

Then,  600  -*-  400  =  1'5,  the  ratio  of  A.  to  BC.  and  of  initial  squares 
at  one  end  to  rectangles  at  the  other. 

And  in  the  3  main  sections  of  the  prismoidal  solid,  Fig.  5, 
We  have  for  similar  sections  of  the  initial  prismoids  = 

Representative.        Dimensions  of  initial  sections.  Initial  areas.       No.      Alain  areas. 

End  A   .  .  .  =  squares  of  1  X  1  .     .     .     .  =  1'       X  400  =    400. 

"    BC    .  .  =  propor.  rectans.  12'5    X  '12  =  \o    X  400  =    600. 

Mid-section .  =        "        "  6'75  X  '56  =  3'78  X  400  =  1512. 

It  will  be  seen  that  the  main  areas  result  as  above  calculated  ; — and 
having  these  and  the  common  length  h.,  it  is  easy  to  compute  the  pris- 
raoid  by  Simpson's  General  Rule,  as  shown  before. 

We  may  add  here,  as  being  indicative  of  the  difficulty  of  comput- 
ing such  a  solid,  by  ordinary  average  rules  (which  answer  tolerably 
well),  in  common  cases. 

That  the  Arithmetical  Mean  of  the  end  areas  =  500,  the  Geomet- 
rical Mean  =  490  ;  while  the  Prismoidal  Mid-section  =  1512,  and 
the  Prismoidal  Mean  Area  =  1174s;  which,  multiplied  by  the  length, 
or  hight,  h.  =  100  :  makes  the  solidity,  above  =  1174662,  or  more 
than  twice  as  much  as  would  result  from  multiplying  the  arithmetical 
mean  by  the  length. 


18 


MEASUREMENT  OF  EARTHWORKS. 


4.  The  Prismoid  adapted  to  Earthwork. — Sir  John  Macneill,  a  dis- 
tinguished English  engineer,  as  early  as  1833,  soon  after  the  intro- 
duction of  railroads,  when  the  necessity  became  apparent  of  having 
ready  and  correct  methods  at  hand  for  computing  the  volume  of  the 
vast  quantities  of  earth,  removed  or  supplied,  in  grading  them,  pre- 
pared and  published  three  series  of  Tables  (in  8vo),  computed  by 
rn^ans  of  the  Prismoidal  Formula.  These  Tables  were  systematically 
arranged,  and  have  been  extensively  used  abroad. 

He  considered  the  Earthwork  Prismoid  as  being  composed  of  a 
Prism,  with  a  wedge  superposed :  since  the  lower  portion  of  the  cross 
section  of  a  railroad,  canal,  or  road  is  generally  symmetrical  and 
regular,  the  ground  surface  alone  being  relatively  variable. 


In  this  diagram  (Fig.  6)  the  reduced  surface  of  the  ground  (taken 
as  level,  crosswise,  or  made  so)  is  shown  by  the  plane  AFGE,  and 
the  cross  section  of  the  road  by  ABCG,  these  are  supposed  to  be 
transparent,  in  order  to  show  the  road-bed  and  mid-section,  as  well  as 
the  far  end  of  the  trapezoidal  prismoid. 

Sir  John  Macneill  commences  his  work,  by  referring  to  a  represen- 
tation of  the  Earthwork  Prismoid  (copied  above),  as  follows : 

"  Let  ABCGFKDE  represent  a  prismoid  or  solid  figure,  similar 
to  that  which  is  formed  in  excavations  or  embankments,  in  which 
BCDK  represents  the  roadway,  and  ABCG,  FKDE,  parallel  cross 
sections  at  each  end.  The  cubic  content  of  this  solid  is  equal  to 


CHAP.  I.— PRELIM.  PROBS.— ART.  4.  19 

The  area  ABCG  -j-  area  FKDE  -f  4  times  area  a  beg, 
Mutipliedby^R: 

"  If,  then,  we  suppose  a  plane,  HIEF,  to  be  drawn  through  the 
lines  HI,  and  EF,  it  will  be  parallel  to  the  base  BCKD,  and  will 
divide  the  solid,  ABCGFKDE,  into  two  others,  one  of  which  will  be 
the  regular  prism,  HBCIFKDE,  and  the  other  will  be  a  wedge,  the 
base  of  which  will  be  the  trapezium,  AHIG,  the  length  IE  or  CD, 
the  length  of  the  prismoid,  and  the  edge  FE,  the  breadth  of  the  cut- 
ting at  the  lower  end  of  the  section." 

The  prismoid,  then,  being  assumed  as  composed  of  a  regular  prism, 
with  a  wedge  superposed,  he  demonstrates  in  the  usual  manner  the 
formula  for  the  volume  of  these  two  solids,  and  shows  that  by  addi- 
tion they  result  in  the  Prismoidal  Formula,  which  he  uses  in  the  com- 
putation of  the  three  series  of  Tables  .which  form  the  bulk  of  his  neat 
octavo  volume  (London,  1833). 

It  will  be  observed  that  all  Macneill's  prismoids  refer  to  ground 
sloping  longitudinally,  but  level  transversely: — to  apply  them,  there- 
fore, to  an  irregular  surface,  it  must  be  first  reduced  to  a  level  cross- 
wise, or  assumed  to  be  so,  practically. 

The  above  extract  from  Sir  John  Macneill's  work  of  1833  is  made, 
not  only  for  its  intrinsic  value,  but  on  account  of  its  being  the  first 
regular  and  successful  attempt  to  adapt  the  Prwmoidal  Formula  to 
the  computation  of  modern  earthworks:  which  is  followed  out  through 
a  series  of  practical  Tables,  comprising  239  pages,  and  extending  to 
50  feet  of  hight  or  depth : — an  embankment  being  considered  as  an 
excavation  inverted. 

This  meritorious  work  of  Sir  John  Macneill  was  speedily  followed 
by  other  writers  in  England,  and  later  by  several  in  this  country.* 
All,  or  most  of  these  productions  being  based  upon  the  Prwmoidal 
Formula  (or  some  modification  of  it),  which  is  now  universally 
acknowledged  to  be  the  only  consistent  and  exact  method  for  com- 
puting the  volume  of  solids  employed  in  modern  earthworks,  and 
even  those  authors  who  employ  pyramidal  rules  are  but  using  a  par- 
ticular case  of  the  former. 

*  Bidder,  Baker^  Bashforth,  Henderson,  Sibley,  Rutherford,  Hughes,  Huntington, 
Law,  Dempsey,  Haskoll,  Morrison,  Rankine,  Graham,  Macgregor,  and  others,  in  England. 
While  in  this  country,  Long,  Johnson,  Borden,  Trautwine,  Gillespie,  Henck,  Davies,  P. 
Lyon,  Cross,  M.  E.  Lyons,  Byrne,  Warner,  Rice,  and  others  (besides  the  present  writer), 
have  dealt  with  this  subject.  Amongst  these,  however,  the  most  comprehensive,  and  the 
best  in  many  particulars,  is  the  work  of  John  Warner,  A.  M.,  a  well  printed  and  hand- 
somely illustrated  8vo,  Philadelphia,  1861,  containing  28  valuable  and  useful  Tables,  and 
14  plates  of  great  importance  to  every  student  of  engineering. 


20 


MEASUREMENT  OF  EARTHWORKS. 


5.  The  Prismoid  in  its  Simplest  Form. — The  unexpected  manner  in 
which  the  Prismoidal  Formula  applies  to  the  cubature  of  other  solids, 
totally1  dissimilar  in  form  and  appearance  (as  to  the  sphere,  taking 
the  poles  as  end  sections  at  zero,  and  the  mid-section  as  a  great  circle), 
justifies  its  consideration  under  various  aspects,  which  would  be 
superfluous  in  any  other  body,  and  hence  we  give  below  a  figure 
illustrating  the  Prismoid,  in  what  may  be  deemed  its  simplest  form 
(when  not  contained  within  a  diedral  angle).  See  Fig.  7,  where  the 
solid  is  level  transversely,  but  sloping  longitudinally,  and  may  be 
supposed  to  represent  (proximately~)  one  of  Button's  Initial  Prismoids, 
square  at  one  end,  and  with  a  proportional  rectangle  at  the  other. 


1 

I 

B 

3 

4 

Fi&7. 

"""*""—  —  -~^__m. 

B 

A. 

-   —  rrrr^fl               

*i                  A.        ^  i 

a. 

6                                                  1 

Here  the  prismoid  is  composed  of  a  prism  on  a  square  base,  with  a 
side  of  1,  and  length  of  6, — and  of  a  wedge,  superposed,  with  a  square 
back,  on  a  side  of  1,  its  edge  also  1,  and  hight  6, — the  common  length 
of  the  two  combined  as  a  prismoid. 

)AA  Represent  the  prism. 
BB  The  wedge. 
m     The  mid-section  of  the  prismoid. 

Then  we  have  for  the  volume  of  this  solid,  by  several  of  the  rules 
already  given. 


Formulas. 


Cubic  ft 


(I.)  (1  X  2)  +  (1  X  1)  +  [(1  +  1)  X  (2  +  1)]  X  -g-  =9 
(II.)  (2  X  2)  +  (2  X  1)  +  (1-5  X  1  X  8)  -4-  12  X  6  .  .  =  9 
(III.)  2  +  1  +  (1-5  X1X4)X-|- -9 

(IV.)  (2X1  +  1  X  2)  +  (2X1  +  1  X  1)  X  |-  •     .  =  9 

Divided  ( Prism    =1X1X6 =6"J 

.=3^=9 


All,  of  course,  resulting  in  the  same  solidity  for  this  simple  pris- 
moid =  9  cubic  feet. 


CHAP.   I.— PRELIM.   PROBS.— ART.   6. 


21 


6.  Further  Illustration  of  MacneilUs  Prismoid. — In  computing  the 
quantities  of  earthwork  for  railroads,  etc.,  it  is  often  useful  (and  gen- 
erally desirable)  to  consider  the  side  slopes,  continued  to  their 
intersection,  above  or  below  the  road-bed  (as  has  been  done  by  T. 
Baker,  C.  E.,*  and  other  writers),  thus  forming  a  constant  triangle 
at  the  intersection,  which  is  deductive  from  the  general  triangular 
figure  formed  by  the  slopes,  and  ground,  in  order  to  obtain  the  regu- 
lar cross  section  of  excavation  or  embankment,  from  ground  to  grade ; 
and  this  triangle  also  forms  the  right  section  of  the  grade  prism,  ter- 
minating the  earthwork  solid  at  edge  of  diedral  angle,  formed  by  the 
side  slope  planes  containing  it. 

To  explain  this  more  clearly,  we  give  a  figure  in  which  both  end 
areas  are  drawn  upon  the  same  plane  (Fig.  8). 

Double  cross  section  of  a  railroad  cut — (in  fact,  Macneill's  pris- 
moid  on  level  ground) — with  road-bed  of  20,  and  slopes  of  1  to  1. 


Prism. 


G^Prism. 


lot  o/ slopes. -10 


References. 

A  =  Altitude  of  grade  triangle. 

B  =  Level  top,  sloping  forward  in  100  feet  to  b. 

b    =  Level  top  of  forward  cross  section. 

G  =  Grade,  or  road-bed,  20  feet  wide. 

c    =  Grade  triangle,  or  constant  end,  of  grade  prism. 

H  —  h  =  Breadth  of  back  of  trapezoidal  wedge. 

r    =  Slope  ratio,  or  in  this  case  1. 

*  Railway  Engineering  and  Earthwork,  by  T.  Baker,  C.  E.  London,  1840.  Wherein 
he  develops  a  very  compendious  and  excellent  system  of  computing  the  earthwork  of 
railways,  which  has  been  extensively  copied. 


22  MEASUREMENT  OF  EARTHWORKS. 

CC  =  Centre  line  of  road. 

I      =  Intersection  of  side  slopes,  or  edge  of  diedral  angle  formed 
by  them. 

To  find  the  equivalent  level  hight — no  matter  how  irregu- 
lar the  ground  may  be. 
Let 

a  =  Whole  area,  to  the  intersection  of  slopes. 

r  =  Slope  ratio. 

h  =  Equivalent  level  hight. 

Then,  \/—  =  h. 
r 

Let  B  and  b  represent  the  level  tops  of  two  cross  sections  of  a  rail- 
road cut,  100  feet  apart  sections,  and  lying  within  the  same  diedral 
angle  of  90°,  formed  by  side  slopes  of  1  to  1,  continued  to  their  inter- 
section, or  edge  at  I. 

Now,  supposing  B  and  b,  to  have  been  originally  a  very  irregular 
surface,  reduced,  by  any  exact  method,  to  the  level  tops  represented. 

Then,  below  b  we  have  a  regular  prism,  .on  a  triangular  base, 
extending  down  to  I ;  and  above  b,  a  regular  wedge  (back  and  edge 
parallel),  upon  a  trapezoidal  back,  of  which  the  base  b  is  equal  to  the 
edge  b,  representing  the  top  of  the  forward  cross  section,  100  feet 
distant. 

Then,  in  the  wedge  above  b,  by  the  properties  of  that  solid,  consid- 
ered as  *  a  truncated  triangular  prism,  and  applicable  either  to  rectan- 
gular or  trapezoidal  wedges, 
We  have, 

Mean  Area. 

(B  +  6-f  6)X(H  — 7<)       (44 '+  32  +32)  X  (22  —  16) 

-g-  -g-  .  108. 

And  in  the  prism  beloiv  b,  down  to  I  (including  the  grade 
triangle) — 

We  have, 

(/j,2  r) =  256. 1 

Deduct  the  grade  triangle =  100.  f  •    •  =  15°- 

Leaves  area  of  prism  (above  grade)  from  G  to  b  =  156. 

Finally,  then,  we  have  the  mean  area  of  the  trapezoidal 
earthwork  solid,  above  grade,  or  road-bed =  264. 

Cubic  Ft. 

Then,  264  X  100  =  26400.     The  solidity  of  this  Prismoid. 

*  Chauvenet's  Gcom.,  vii.  22  (1871),  easily  reducible  to  the  text. 


CHAP.  I.—  PRELIM.   PROBS.—  ART.  6.  23 

If  more  convenient,  we  might  exclude  entirely  the  grade  triangle, 
and  stop  the  calculation  at  G  (the  road-bed),  but  as  a  system  of  com- 
putation, and  in  view  of  the  simplicity  of  the  geometrical  relations  of 
triangles,  it  will  usually  be  found  best  to  include  the  grade  triangle 
as  above,  and  ultimately  to  deduct  it,  in  some  form. 

The  employment  of  the  method  of  this  article  enables  us  to  find  a 
mean  area  to  the  prismoid  —  without  using  a  mid-section  —  and  this 
mean  area,  when  multiplied  by  the  length,  gives  the  volume  of  the 
whole  solid. 

Thus  we  may  assume  any  level  trapezoidal  prismoid  of  unequal 
parallel  ends  (as  Macneill  does),  to  be  composed  of  two  solids  —  a 
prism,  with  a  wedge  superposed. 

1.  A   Triangular  Prism,  with  a  cross  section,  equivalent  to  the 

lesser  end,  supposing  the  slopes  to  intersect,  and  embracing 
the  grade  triangle. 

2.  A  Trapezoidal  Wedge,  superposed  upon  the  prism,  having  an 

area  of  back  equivalent  to  the  difference  of  the  ends,  its 
edge  being  the  level  top  of  the  smaller,  and  equal  to  the 
base  of  the  back. 

The  length  being  common  to  both  partial  solids,  and  to  the  whole 
prismoid. 

Then,  for  the  mean  area  of  the  wedge,  we  have, 
(B  +  &  +  ft)  X  (H  —  A)* 

6 

and  for  that  of  the  prism  to  intersection  of  slopes  —  (A2  r  —  grade 
triangle),  and  by  addition  ,f 


the  common  length  =  The  Solidity  of  the  Prismoid  ....     (VI.) 

Or,  in  words,  —  The  sum  of  the  mean  areas  of  the  prism,  and  super- 
posed wedge,  multiplied  by  the  common  length,  equals  the  solidity  of  this 
prismoid. 


*  Chauvenet's  Geom.,  vii.  22  (1871). 

f  B  and  b  are  always  the  widths  between  top  slopes  at  the  ends. 

And  H  —  h  (however  irregular  the  ground  line  of  the  ends  may  be)  is  obtained  by 

dividing  the  difference  of  end  areas  by  half  the  sum  of  their  top  widths,  or  (     +    V 
See  note  at  foot  of  this  Article  6. 


24  MEASUREMENT   OF   EARTHWORKS. 

Note. — When  the  ground  surface,  or  upper  side  of  the  superposed 
wedge,  is  very  irregular  (as  in  Figs.  43  and  44) —  ascertain  the  hori- 
zontal widths  of -each  end  at  top  slope.  Then  the  difference  between 
the  areas  of  the  two  ends  is  the  surface  of  the  back  of  the  superposed 
wedge,  and  this,  divided  by  the  average  of  the  two  horizontal  widths 
above,  gives  the  vertical  hight  of  the  back,  or  altitude  of  the  trian- 
gular section,  of  which  the  length  of  the  prismoid  is  the  base,  giving 
at  once  the  means  of  computing  its  area,  and  this,  multiplied  by  one- 
third  of  the  sum  of  the  lateral  edges,  gives  the  solidity  of  the  superposed 
wedge.  (Chauvenct,  Geom.,vii.  22.) 

7.  Trapezoidal  Prismoid  of  Earthwork,  considered  as  two  Wedges.— 
On  ground,  either  level  crosswise,  or  reduced  to  an  equivalent  level 
by  any  correct  process,  an  Earthwork  Prismoid,  within  the  limits  of 
its  slopes,  road-bed,  and  ground  surface,  may  readily  be  computed  aa 
two  wedges  (Hutton's  Particular  Rule),  without  an  assumed  mid-sec- 
tion, or  even  the  end  areas. 

And  in  this  there  is  some  advantage,  as  the  width  of  road-bed  at 
the  end  sections  may  be  unequal  to  any  extent,  provided  the  widening 
is  gradual. 

Thus,  let  Fig.  9  represent  a  regular  station  of  a  railroad  cut,  100 
feet  in  length,  with  slopes  of  1  to  1,  and  in  the  near  end  section  a 
depth  of  40  feet,  and  road-bed  of  20,  while  in  the  far  one  it  has  a 
depth  of  30,  and  road-bed  of  40  feet  wide. 

Hutton's  Particular  Rule,  modified  for  application  to  earthwork, 
may  be  expressed  in  words  at  length  as  follows : 

Rule. 

Add  road-bed  -f-  top  width  +  road- 

r    .,  .  ,.        I   bed  of  2d  section;  multiply  the  sum 

In  1st  cross  section   -<  v    i      i    £•  i       ^ 

of  these  three  by  level    hight  of  sec- 
tion, and  reserve  the  product. 

Add  road-bed  -j-   top   width  +    top 

T    m  I   width  of  1st  section  ;  multiply  the  sum 

In  Id  cross  section   «     -  xl-  ,      ,      ,    , \  *_      „ 

of  these  three  by  level   hight  of  sec- 
tion, and  reserve  the  product. 

Finally,  add  the  two  products  reserved,  and  i  of  their  sum  is 
the  mean  area  of  the  Prismoid,  which,  multiplied  by  length  = 
Solidity (VII.) 


CHAP.   I.— PRELIM.    PROBS.— ART.   7. 


25 


Eeferring  to  Fig.  9,  the  line  CC  is  the  centre  line  traced  upon  the 
ground,  and  below  it  the  road-bed  gradually  widened  from  20  to  40 
feet,  in  the  length  of  100 ;  the  figures  marked  show  the  dimensions 
assumed  for  illustration,  and  the  dotted  lines  the  edges  of  a  plane 
supposed  to  be  passed,  so  as  to  convert  this  solid  into  two  wedges. 

The  nearest  having  a  trapezoidal  back,  standing  on  a  road-bed  of 
20,  with  a  hight  of  40,  and  its  edge  being  the  road-bed  of  40  feet 
wide,  belonging  to  the  far  cross  section. 

The  farthest  wedge,  above  the  dotted  lines,  having  for  its-  back  the 


far  section,  standing  on  a  road-bed  of  40,  with  hight  of  30,  and  its 
edge  being  the  top-width  of  the  near  cross  section,  100  feet  wide,  at 
ground  line. 

[In  Chapter  5  we  shall  consider  further,  and  more  in  detail,  the 
subject  of  Wedges  ;  and  their  application  to  the  computation  of  earth- 
work solids,  and  illustrate  it  by  several  examples.  Comparing  also 
the  results  obtained  with  those  derived  from  the  use  of  BUTTON'S 
General  Rule: — which  is  the  accepted  standard  for  accuracy  in  such 
work.] 


26 


MEASUREMENT  OF  EARTHWORKS. 


EXAMPLE. 


By  Our  Modification  of  Hutton's 
Rule (VII.) 


In  1st  cross  section 


In  2d  cross  section 


20 
100 
40 

160 
40 

6400 

40 
100 
100 

240 
30 

7200 

6400 
7200 

6)13600" 


Mean  Area  = 


2266-67 
100 


By  Button's  Particular  Rule.  (IV.) 

Reducing  Trapezoids  to  Rectangles. 

Mean  breadths  = 


60 

2 

120 
40 

160 
40 


70 

2 

140 
100 

240 
30 


6400        7200 

6400 

7200 


Solidity  .  . 


13600 
100 

6)1360000 
226667 


Solidity  .   .  =      226667'00 


8.  Areas  of  Railroad  Cross-sections  (within  Diedral  Angles] — 
whether  Triangular,  Quadrangular,  or  Irregular. 

All  railroad  sections  are  contained  within  diedral  angles,  formed  by 
side  slope  planes,  of  a  given  divergency— determined  by  the  slope 
ratio  (r). — The  edge  of  this  diedral  angle  is  a  right  line,  parallel  to 
the  grade,  and  prolonged  forward  indefinitely  from  I,  the  intersection 
of  the  side  slopes  (in  a  right  section),  until  the  end  of  the  cut  or  fill 
is  attained.  Here,  at  the  grade  point,  it  changes  its  position  to  a 
corresponding  parallel  above,  or  below,  as  the  case  may  be.  Consid- 
ering, with  Sir  John  Macneill,  an  embankment  to  be,  in  effect,  an 
excavation  inverted,  the  situation  of  the  edge  of  the  diedral  angle,  or 
intersection  of  the  slopes,  will  generally  (in  our  examples)  be  found 
below  the  road-bed,  but  always  parallel  to  the  grade  line,  and  at  the 
same  distance  from  it,  as  long  as  the  side  slopes  continue  uniform. 

(a.) From  the  geometrical  relations  of  triangles  and  rect- 
angles, it  is  obvious  that  in  a  triangle  situated  as  in  Fig.  10 — con- 


CHAP.    I.— PRELIM.    PROBS.— ART.    8. 


27 


tained  within  rectangular  axes  and  their  parallels,  and  divided  into 

two  by  the  central  axis  h,  the  area  of  the  whole  is  equivalent  to  — 

2i. 

—  the  parallels  a  and  b,  to  the  centre  line  h,  limiting  the  triangle 
laterally. 

The  same  rule,  precisely,  applies  to  quadrangles,  which  may  always 
be  cut  by  a  diagonal  into  two  triangles. 

This  rule  (in  fact),  equally  applicable  both  to  triangles  and  trape- 
ziums, is  that  laid  down  by  Hutton  (1770)  for  trapeziums. 

In  Fig.  10, — h  X  w  =  double  area  of  the  whole  triangle,  whose  ver- 
tex is  at  I,  the  intersection  of  the  slopes,  and  its  sides,  the  side-slopes, 
and  the  ground  line.  Thus,  let  h  =  20,  w  =  45,  then  20  X  45  = 
900  -f-  2  =  450,  area  of  whole  triangle ;  but  it  is  often  more  conve- 


nient, in  calculations,  to  use  double  areas  alone,  until  the  close  of  the 
operation,  as  in  many  problems  of  land  surveying. 

In  a  triangle,  the  direct  axes  h  or  h'  may  take  any  position,  pro- 
vided the  parallels  through  the  lateral  vertices  are  made  to  follow, 
and  the  tranverse  axes,  w  and  w',  remain  rectangular. 

But  in  a  quadrangle,  the  position  of  the  direct  axis  is  fixed  by  that 
of  the  opposite  vertices,  through  which  it  passes,  and  with  it  the  axis 
of  width,  and  its  limiting  parallels,  are  also  fixed. 

In  Fig.  10,  suppose  the  direct  axis  and  its  parallels  to  revolve  upon 
I,  into  the  position  h',  and  that  h'  becomes  22*1  —  then  it  will  be  found 

thatuf  has  become40'73, 


will  be  22>1  X  40'73  «  450j 


area  of  whole  triangle,  as  before. 


28 


MEASUREMENT  OF  EARTHWORKS. 


In  both  these  cases,  Figs.  10  and  11,  each  figure  is  divided  by  the 
centre  line,  or  direct  axis,  into  two  triangles,  having  a  common  base, 
and  contained  between  parallels  to  it,  drawn  through  the  opposite 
vertices. 

In  both  Figs.  10  and  11,  h  X  w  =  double  area  of  the  figure  to 
which  they  relate, — as  these  are  rectangular  factors,  for  determining 
the  content  of  the  wholly  or  partially  circumscribing  rectangles 
(between  the  same  parallels),  of  which  the  triangle  or  trapezium 
represented,  is  each  equivalent  to  one-half. 

This  rule  is,  in  fact,  the  simplest  possible,  being,  substantially,  the 
definition  of  a  plane  surface,  length  X  breadth  (which  indicates 
superficial  extension),  and  from  its  extreme  simplicity,  there  seems  to 


be  no  adequate  reason  why  it  should  not  be  more  generally  employed, 
for  although  its  application  to  ^triangular  surfaces  necessarily  gives 
double  areas, — a  division  by  two  is  the  briefest  imaginable. 

Right  and  left  of  centre  each  triangle  is  obviously  equal  to  half  the 
rectangle  of  the  hight  and  width  on  that  side  (the  triangle  and  rect- 
angle having  a  common  base,  and  lying  between  the  same  parallels, 
a  and  b),  and  by  addition,  the  double  area  of  the  whole  trapezium  = 
hight  X  width. 

(b.) In  view  of  the  rule  just  recited,  for  finding  the  areas 

of  triangles  and  trapeziums,  by  hights  and  widths,  it  becomes  of  some 
importance  to  have  a  concise  rule*  for  determining  the  distances  out 
of  the  vertices  from  the  axis,  when  the  hight  and  slopes  alone  are 

*  GHlespie,  Roads  and  Railroads  (1847),  gives  rules  analogous  to  ours,  but  they  had 
long  before  been  kiwion. 


CHAP.  I.— PRELIM.  PROBS.— ART.  8. 


29 


given :  in  this  there  is  little  difficulty,  as  engineers  have  long  been 
possessed  of  formulas  for  the  purpose,  similar  to  those  which  will  be 
seen  below,  referring  to  Figs.  12  and  13, — and  these  distances  out,  when 
added  together,  form  the  width  w,  of  the  rule  above. 


In  Fig.  12. 


Ht.  TTid. 

40  X  60-8 


2432 
2 


Area. 

1216. 


Both  in  trapeziums  and  triangles  the  diagonal  X  the  sum  of  per- 
pendiculars from  the  opposite  angles  =  double  area. 
Or,  centre  Light  X  the  total  width  =  double  area. 


Suppose,  in  both  these  figures,  the  side-slopes,  ground-slopes,  and 
centre  hight,  or  axis,  given,  and  the  side-slopes  intersected  at  I,  then 
to  find  the  distances  out,  right  and  left  of  cejitre,  take  each  side  sepa- 
rately. Consider  the  centre  line,  or  axis,  to  be  a  meridian  (as  in  a 
map),  imagine  also  an  east  or  west  line,  drawn  through  the  origin  of 
each  slope  (side  or  ground). 

Then, 

If  the  slopes  incline  towards  the  same  compass  quarter  : 

Hight 

^5 — jr& j —  — — p-j =  distance  out  =  cL 

By  difference  of  nat.  tans,  of  slopes 

If  the  slopes  incline  towards  adjacent  compass  quarters: 

Hight 

^—        -7—         F~l =  distance  out  —  d. 

Uy  sum  oi  nat.  tans,  of  slopes 

These  results  on  both  sides  of  centre,  added  together,  give  the  total 
width  of  the  whole  trapezium. 


30 


MEASUREMENT   OF  EARTHWORKS. 


In  Fig.  13. 

Ht.  Wdt.  Area. 

30  X  88-2       2646 
2         "     2     * 


1323. 


These  rules  also  furnish  a  concise  and  easy  method  of  finding  the 
half  breadths,  a  matter  deemed  quite  important  by  foreign  engineers. 

(C.) The  side  slopes  (bounding  the  diedral  angle)  remain- 
ing plane  surfaces  as  usual  in  the  cross-sections  of  earthwork,  we 
sometimes  find  the  ground  surface  very  irregular,  but  even  these 
cases,  upon  the  principle  of  equivalency,  may  be  correctly  dealt  with, 
so  as  to  reduce  them  easily  to  the  plane  figures  of  the  elements  of 
geometry. 


Thus,  although,  as  far  as  we  have  shown,  the  rule  of  — ,  applies 

only  to  a  line  once  broken,  so  as  to  change  the  figure  considered,  from 
an  oblique  triangle  into  a  trapezium ;  nevertheless,  it  is  not  difficult 
to  reduce  or  equalize  a  surface  line,  very  much  broken,  by  a  single  one 
properly  drawn,  which  shall  contain  within  it  an  area  exactly  equal  to 
that  bounded  by  the  irregular  outline,  and  thus  bring  it  within  the 
rule. 

In  Fig.  14,  let  ABCDEFGH  be  the  cross-section  of  a  rail- 
road cut,  base  20,  slopes  1  to  1,  intersecting  at  I,  the  centre  line  being 
marked  CC — (this  area  looks  irregular  enough,  but  had  it  been  ten 
times  more  so,  the  process  below  would  have  equalized  it  exactly.') 

Then,  from  the  top  of  the  shortest  side  hight  at  H  (adopted  for 
convenience),  draw  a  line  HK  parallel  to  the  road-bed,  or  base  AB, 


CHAP.   I.— PRELIM.   PROBS.— ART.   8. 


31 


making  a  level  trapezoid  10  feet  high  upon  the  section,  or  ABKH  = 
300  in  area. 

Now,  we  will  find,  by  a  common  calculation,  the  area  of  the  whole 
cross-section — between  base  AB,  side  slopes,  and  broken  ground  line 
— to  contain  =  654  area.  Neglecting  in  this  case  the  grade  triangle  at 
I,  as  being  a  common  quantity,  not  affecting  the  result : — (but  adding 
the  grade  triangle  (100),  the  area,  from  the  ground  line  down  to  the 
edge  of  the  diedrul  angle  at  I  =  754). 

Then,  654  —  300  =  354,  the  area  of  the  partial  cross-section  above 
UK,  extending  to  the  irregular  outline,  which  is  to  be  correctly  equal- 
ized,  by  a  single  sloping  line  drawn  from  H. 


40 

7- 


Clrcum:Gr: 
«n-S 

Area  I 


*Tr*  xT!al:Ia.Cir. 


Now,  =  17-7 


LM,  the  altitude  of  a  triangle  HKM,  on 

the  base  HK,  which  is  exactly  equivalent  in  area  to  the  partial  cross- 
section  above  HK. 

So  that  HM  is  a  single  equalizing  line,  drawn  from  H,  equivalent 
to  the  broken  line  of  ground,  and  including  the  same  area  exactly. 
Another  way  of  finding  the  point  M  —  the  terminus  of  the  equaliz- 

f  Double  area  =  1508       IM 
ing  line— is  the  following :  \ =  53'3  ]• and 

(      IHXsin.ofI 
this  is  a  very  concise  method,  as  IH  is  easily  found.* 


VI) 

4,, 


*  This  rule  will  be  found  useful  as  a  verification  of  the  process  of  Fig.  14. 


32  MEASUREMENT  OF  EARTHWORKS. 

If  the  degree  of  equivalent  surface  slope  be  desired  (as  it  usually  is), 

Then,  ^-  =  cot.  17°  (nearly)  =  3'26. 

The  slope  of  the  equalizing  line  HAI  being  17°  ascending  from  H, 
we  easily  find  FN  =6'135,  and  adding  FI  =  20,  we  have  IN  or  h  = 


26-135,  and  w  =  57'7.     Then, 


h  X  iv  =  26-135  X  57-7 


754,  and 


deducting  the  grade  triangle  (ABI  =  100),  we  have,  finally,  the  area 
of  the  whole  cross-section  above  the  road-bed  =  654,  thus  verifying 


the  original  calculation  as  before  given,  and,  by  using  the  radii  of 
inscribed  and  circumscribed  circles,  we  can  prove  it,  if  necessary : 

(Fig.  14). 

(d.) It  is  sometimes  desirable,  by  means  of  an  equalizing 

line,  to  deal  with  the  boundary  alone,  without  the  rest  of  the  cross- 
section,  and  this  is  not  difficult,  for  we  may  consider  the  broken  line 
HKM  (Fig.  14),  or  aeg  (Fig.  15),  as  a  base  of  ordinates,  preserving, 
however,  their  parallelism,  and  taking  all  the  distances  horizontally 
as  though  the  base  were  straight  (see  Fig.  15)  ;  but  the  process  of  Fig. 
14  is  generally  preferable. 


CHAP.   I.—  PRELIM.   PROBS.— 


, 

Q&  U  ^ 


It  is  often  useful  to  equalize  a  section  by  a  level  top  line,  or  slope 
o/0°.     This  can  be  done  as  shown  in  Art.  6. 

Whole  area =  a. 

Slope  ratio =  r. 

Level  hight =  h. 


Then  h 


The  ordinates  marked  upon  Fig.  15  are  deduced  from  those  of 
Fig.  14,  and  the  calculations  of  the  irregular  area,  a  eg,  are  made  by 
successive  trapezoids,  and  double  areas,  as  follows : 

Ordinates      in  f  a  _}_  J      b  -\-  C      C  +  d      d  +  e         6      +/         /     +^ 

Ease3  Hne7ai«!l  °  +  5      5  +  2      2  +  6       6  +  16       16+16       16  +  0 
broken  at  e.  .  .  .  (       5  7  8  22  32  16 

10          10          10  10  4  10 

50+70    +    80+220+128      +      160 

Then,* 

Sum  of  double  areas  =  708 

T5 1- r^ = \ —          -77.  =  1  r7  =  ft  A:,  as  before. 

.base  or  equalizing  triangle,  a  e  =  40 

And  ak  is  the  equalizing  line,  ascending  from  a,  with  a  slope  of 
17°,  which  is  equivalent  to  HM,  of  Fig.  14. 

(6.) We  may  now  briefly  refer  to  the  computation  of  cross- 


Horizontal  distances 
•part 


Double    areas    (total 
70S) 


sections.     These  are  usually  taken  in  the  field  with  the  rod,  level,  and 
tape;  they  designate  by   levels,  and   distances  out,  the  prominent 

*  With  equal  abscisao,  Simpson's  well-known  rule,  or  that  of  Davies  Legendre,  would 
conveniently  apply. 

3 


34  MEASUREMENT   OF  EARTHWORKS. 

points,  or  features  of  the  ground,  and  fix  the  intersection  of  the  side 
slopes,  or  place  of  the  slope  stake,  which  bounds  the  limits  of  excava- 
tion or  embankment ;  and  on  regular  ground,  the  clinometer  may  be 
used,  but  is  less  correct  and  satisfactory. 

On  plain  ground,  but  three  levels  are  taken, — the  centre  and  side 
hights, — and  this  has  been  called  three-level  ground.  It  is  the  prac- 
tice of  many  engineers  (and  it  is  a  good  one)  to  take  angle  levels  and 
distances  over  the  edges  of  the  road-bed,  this,  then  becomes  five-level 
ground;  and  where  more  than  five  levels  are  necessarily  taken,  the 
cross-section  is  usually  deemed  irregular,  though  the  point  where 
sections  become  irregular  is  not  well  defined,  and  may  be  safely  left 
to  the  judgment  of  the  engineer. 

In  this  case  (Fig.  16),  the  centre  and  side  hights,  and  the  right  and 
left  distances  out  to  the  slope  stakes,  are  always  given,  and  the  calcu- 
lation becomes  simple  and  rapid. 

The  following  is  the  method  long  ago  used  by  engineers,  and  pub- 
lished by  Trautwine  *  and  others,  twenty  years  since. 

KULE  for  area  of  cross-section,  with  uniform  road-bed  and  centre 
and  side  hights  given. 

Half  the  centre  cutting  X  by  right  and  left  distance,  plus  right 
and  left  cuttings  X  one-fourth  of  road-bed. 

Thus,  in  Fig.  16, 


We  have,  by  this  rule, 

5  X  64  =  320. 
44  X    5  =  220. 

Area.  .  =  540. 


And  by  using  the  grade  triangle  and 
hights  and  widths,  as  in  Figs.  10  and  11, 


We  have, 


20  X  64 


.=640. 


w  —  64.  J  Less  grade  triangle  .  =  100. 
Area.  .  =  54a 


(f.) To  find  the  area  of  cross-sections,  where  angle  levels 

have  been  taken,f  or  Jive-level  ground  (which  angle  levels  have  long 
been  used  by  engineers,  and  are  recommended  by  Prof.  Davies  in  his 
new  surveying),  we  will  give  an  example  for  illustration,  from  which 
the  rule  of  this  method  will  be  evident.  (See  Cross,  Eng.  Field 
Book,  N.  Y.,  1855.) 

*  Trautwine's  New  Method  of  Ex.  and  Em.  (1851). 

f  Davies'  New  Surveying  (1870), — cross-section  levelling. 


CHAP.  I.— PRELIM.  PROBS.— ART.  8. 


35 


Now,  to  calculate  the  area  of  this  cross-section,  Fig.  17,  by  double 
areas, 

Equivalent  to, 
Triangle,    15  X  10 


We  have, 

By     divid- 

20 X  15  =    300. 

ing  the  figure 

20  X  12  =    240. 

into  six  trian- 

34 X  16  =    544. 

gles,  or  three 

2)1084. 

trapeziums. 

Area.  =    542. 

Trapezoid,  27  X  10 


Triangle, 


28  X  10 
16  X  24 


=  150. 

=  270. 

=  280. 

=  384. 


2)1084. 


Area. 


=    542. 


To  compute  this  area  in  the  usual  method  by  successive  trapezoida 
and  deductive  triangles,  is  much  longer  and  less  satisfactory. 


-i<* 


(g.) For  very  irregular  cross-sections,  no  definite  rule  can 

be  given, — they  are  usually  reduced  to  elementary  forms,  which,  being 
separately  computed,  and  finally  totalized,  give  the  whole  area  in 
the  end. 

This  reduction  is  usually  made  to  trapezoids  and  triangles  (additive 
or  deductive],  while  the  calculations  are  the  simplest  possible,  though, 
from  the  multitude  of  figures,  necessarily  tedious. 

In  the  most  irregular  sections,  involving  heavy  rock-work  on  side- 
hill, — the  several  cuttings  (or  level  hights),  transversely,  are  fre- 
quently taken  at  ten  feet  only,  or  some  such  uniform  distance  apart, 
and  in  these  cases  the  mean  hights  of  a  number  of  contiguous  trape- 
zoids may  be  ascertained,  and  multiplied  by  the  uniform  distance 
(agreeably  to  the  rules  of  mensuration  for  irregular  areas),  and  thus 
abbreviate  somewhat  the  labor  of  such  computations ;  which,  how- 
ever, in  their  origin,  and  indispensable  verifications,  are  often  laborious 
enough,  though,  fortunately,  so  simple  and  elementary  as  to  be  within 
the  comprehension  of  all  the  members  of  an  engineer  party,  which 
enables  us  to  bring  many  hands  to  the  work. 


36 


MEASUREMENT  OF  EARTHWORKS. 


Not  nnfrequently,  too,  in  rock -work  (proximating  a  cost  of  a  dollar 
per  cubic  yard),  it  has  been  deemed  necessary  to  take  independent 
cross-sections,  at  only  ten  feet  apart  forward,  over  the  roughest  por- 
tions of  the  work. 

In  that  event,  although  the  calculations  become  voluminous,  we 
have  the  satisfaction  of  knowing  that  the  solidity  is  correctly  obtained  ; 
since,  in  such  short  spaces,  no  ordinary  rules  would  produce  any 
important  variation  in  the  final  result ;  supposing,  of  course,  the 
cross-sections  to  be  correctly  laid  out,  and  measured  with  accuracy, 
both  horizontally  and  vertically — a  matter  of  no  small  difficulty  on 
steep,  rocky  hill-sides,  when  cleaned  for  ivork. 

9.  Further  Illustration  of  the  Modification  of  Simpson's  Rule — (II.)> 
with  a  Diagram  Representing  it,  and  also  one  of  the  Regular  Formula, 
and  another  Modification. 

Here  let  us  take  the  triangular  prismoid,  cross-sectioned,  in  Fig.  8 
(and  shown  below),  and  suppose  its  length  100  feet  (A) — the  end 


Tig  18. 


e.---* 


cross-sections  being  dimensioned  as  before.  With  road-bed  of  20,  and 
slopes  of  1  to  1.  The  whole,  shown  in  projection,  to  give  a  better 
idea  of  the  nature  of  the  solid. 


CHAP.   I.— PRELIM.    PROBS.— ART.   9.  37 

References. 

CC         =  Centre  line  and  edge  diedral  angle. 

ACCB  =  Grade  prism. 

AB        =  Road-bed,  20. 

AE        =  Side-slope  plane,  1  to  1. 

EF        =  Ground  plane,  assumed  as  level. 

ea&E  =  Wedge  of  Fig.  8. 

Then,  for  the  volume  of  this  solid,  we  have,  by  the  modification  of 
Simpson's  Rule  (II.), 

Ilightg.  Widths. 

Near  end  (double  area),  22  X  44     .     .     .  =    968  =  2  b. 
Far  end,  "  16  X  32   '.     .     .  =    512  =  2*. 

8  times  mid-section,  .     .  38  X  76  1 

. ,      >  =•  zooo  =  o  in. 
=  sum  hts.  X  sum  wids.  j 

12)4368 

Mean  area.  .     .  =*    364 
Length  h.     .     .  =         100 
Whole  triangular  solid  to  intersection  ) 

of  slopes.    .' / 

Deduct  grade  prism  under  road-bed.     .  .  =     10000 

Leaves  volume  above  road-bed,  or  Trape-  ) 

-j  i  r>  •       -j    f  77-    ti       7  r  —     20400  =   The  same 

zoidal  Prismoid  of  Earthwork.     .     .  j 

solidity,  as  before  computed,  Art.  6. 

(a.) The  transformation  or  modification  of  Simpson's  Rulo 

(II.)  may,  in  its  mid-section  term,  be  conveniently  represented  by  \\ 
diagram  (perhaps  more  curious  than  useful). — Thus,  continuing  the 
side-slopes  through  the  intersection,  so  as  to  form  the  end  cross-sec- 
tions, one  above  the  other. 

So,  in  Fig.  19,  dimensioned  as  in  Fig.  8,  we  have, 

The  triangle    IEF        =  The  larger  end  section,  or  area. 

"          "         ICD        =  The  smaller  one. 

;  .         "    rectangle  KLMN  =  8  times  the  area  of  the  mid-section, 

or  the  circumscribing  rectangle 
formed  by  sunn  of  hights  X  sum 
of  widths. 

The  road-beds  .     .     .  =  The   dotted    lines,   and    may  be 

assumed  (parallel)  anywhere. 


38 


MEASUREMENT  OF  EARTHWORKS. 


The  parallelogram  IFEP  =  Higlit  X  width  of  larger 

end,  or  double  area  of  .    A. 
IDCO  =  Hight  X  width  of  smaller, 

or  double  area  of.     .     .     B. 

"  rectangle        KLMN  =  HG  X  OP,  or  sum  Lights  X 

sum  widths,  =  8  times  the 
mid-section. 

Here  it  is  evident  that  IH   X   FE  =  Double  area  of  larger  end 
section,  or  =  IFEP  ......  and  IG   X   CD  =  same  of  smaller  = 

IDCO. 


While  (CD  +  FE)  X  (GI  +  IH)  =  the  circumscribing  rect- 
angle KLMN  =  HG  X  OP,  or  the  rectangle  of  sum  of  hights  and 
sum  of  widths. 


Also, 

/HI  -f-  IG\      "/FE  +  CD^ 


19 


— =  861,  the  mid-sec. 


<  \        2        /  /N  V        2 
(  HG  X  OP,  or  38  X   76 =  2888,  or  8  times  mid-sec.  : 

The  triangles  Q  and  R  taken  together  =  the  Arithmetical  Mean  of  A 
and  B,  the  end  areas  =  (16  X  8)  -f  (22  X  11)  =  128  +  242  =  370,  or 

484+256       740  ,  .,.     4.    ,  lf 

— j- =  — —  =  370,  the  Arithmetical  Mean. 


CHAP.   I.— PRELIM.    PROBS.— ART.    9.  39 

The  triangles  T  and  T  are  each  equal  to  the  Geometrical  Mean  of 
the  end  sections  A  and  B  =  \^484  X  256  =  352. 

While  U  and  V  added  together  proximately  equal  the  Harmonic 
Mean  between  A  and  B,  or  =  334. 

So  that  the  circumscribing  rectangle,  KLMN,  representing  the 
mid-section  term,  of  Simpson's  Transformed  Rule  (II.)*  contains,  or  is 
composed  of,  the  following  areas. 


Double  area  of  A. 
"        "       B. 


C  484 
'  \  484 
f  256 
•    •  (  256 

(The  two  end  sections.) 
Arithmetical    Mean  ......    370 

Geometrical  Mean  X  2.     .     .     -{352 
Harmonic   Mean  .......    334 


Total  8  times  the  mid-sec., 
or  361  X  8.  =  2888 


In  this  case :  0 
=  Double  areas 
of  both  ends  + 
4  times  the  Geo- 
metrical Mean 
=  2888. 


Some  curious  inferences  may  be  drawn  from  this  diagram,  but  their 
practical  results  can  be  more  concisely  obtained  in  other  forms. 


Diagram  of  the  regular  Prismoidal  Formula  of  Simpson  and  Hutton. 

As  applied  to  a  triangular  prismoid,  formed  by  a  diagonal  cutting 
plane,  from  the  rectangular  prismoid,  Fig.  2,  and  shown  again  in  Figs, 
22,  24,  and  52,  with  side-slopes  of  li  to  1. 


40  MEASUREMENT   OF  EARTHWORKS. 

Let  1  (Fig.  19*)  Be  the  larger  end  section  (Fig.  22),  transformed 

into  an  equivalent  right  triangle. 

3  The  smaller  end  (Fig.  24),  also  transformed  :—  4  and  5, 
additive  triangles,  making  up  the  trapezium  ABCD  (Fig. 
19  J),  equivalent  in  area  to  four  times  the  prismoidal  mid- 
section  (Fig.  23). 

From  this  diagram  we  readily  deduce  a  simple  modification  of  the 
prismoidal  formula,  equivalent  in  remit,  for  triangular  prismoids. 

Higlits.  Widths. 

-P..  «  f  h  =  30  X  90  =  w 

Dimensions  of  <% 

Figs.  22  and  24.  J   .  /\ 

j   K  =  20  X  60  ==  w' 

(^  Length  =  100,  usually. 


Then,  -  —  -  —  —  x  length  =  Solidity.   .   VIII, 

This  operates  very  simply  in  figures,  by  direct  and  cross  multiplica- 
tion of  hights  and  widths. 

Substituting  the  numbers,  Solidity  —  95000,  as  hereafter  computed, 
Art.  10  (a). 

10.  Adaptation  of  the  Prismoidal  Formula  to  the  Quadrature  and 
Cubature  of  Curves,  and  also  Solids,  where  the  Ordinates  are  equivalent 
to  Sections  —  by  the  Method  of  Simpson,  as  explained  by  Hutton. 

The  eminent  mathematician,  THOMAS  SIMPSON,  to  whom  we  are 
indebted  for  the  Prismoidal  Formula,  also  devised  a  method  for  the 
quadrature  of  irregular  curves  by  'means  of  equidistant  ordinates,  or 
for  their  cubature,  by  using  equivalent  sections  of  irregular'solids,  at 
equal  distances,  instead  of  ordinates  ;  such  solids  being  bounded  oppo- 
site the  base  by  a  general  curved  outline. 

This  method,  although  a  century  old,  is  still  the  simplest  and  best 
yet  known  for  proximating  the  area  of  irregular  curves,  or  the  volume 
of  unusual  solids,  —  it  has  attained  great  celebrity,  and  been  of  much 
service  to  philosophers  and  calculators,  ever  since  its  origin  in  1750. 

It  has  long  been  used  by  military  engineers  for  ascertaining  the 
volume  of  warlike  earthworks,  and  is  regularly  quoted  in  the  leading 
text  books  of  that  important  profession.* 

Also  by  naval  architects  in  determining  the  nice  problem  of  the 
displacement  of  ships  ;  by  mechanical  philosophers,  like  Morin  and 

*  LaisnS,  Aide  Memoire,  du  G6nie.—  Eds.,  1831-61. 


CHAP.   I.— PRELIM.   PROBS.— ART.   10.  41 

Poncelet,  etc. — by  these  it  has  been  deemed  of  much  importance,  not 
only  for  the  quadrature  uf  irregular  areas,  but  also  for  the  ''Cubature 
of  solids  of  irregular  excavations,  embankments,  etc."  * 

It  forms  a  leading  feature  in  Button's  remarkable  chapter  on  the 
cubature  of  curves  (who  seems  to  have  fully  adopted  it),  under  the 
name  of  the  method  of  equidistant  ordinates. — (See  4to  Mens.,  1770, 
sec.  2,  part  iv.  page  458.)— We  are  much  indebted  to  Hutton  for  the 
practical  development  of  this  important  problem,  and  he  gives  several 
examples  of  its  utility.  Amongst  others,  computing  the  area  of  a 
quadrant  of  a  circle,  with  radius  =  1, — which,  by  Simpson's  method, 
using  11  ordinates,  gives  *7817  area,  instead  of  '7854— "pretty  near 
the  truth  "  (says  Hutton). 

"We  will  describe  this  method  from  the — (4to  Mens.,  1770,  p.  458). 

"If  any  right  line,  AN,  be  divided  into  any  even  number  of 

equal  parts,  AC,  CE,  EG,  etc.,  and  at  the  points  of  division  be 

erected  perpendicular  ordinates,  AB,  CD,  EF,  etc.,  terminated 

by  any  curve,  BDF,  etc." 

Then,  the  sum  of  the  first  and  last  ordinates,  plus  4  times  sum  of 
even  ordinates,  plus  2  times  sum  of  odd  ones,  -f.  by  3,  and  X  by  AC, 
one  of  the  equal  parts ;  the  resulting  product  will  equal  the  area, 
ABON,  "very  nearly." 

That  is  to  say,  if 

The  sum  of  the  two  extreme  ordinates  .     .  =  A.   |       ,-, 

"       of  all  the  even  numbered      "       .     .  =*  B. 

f  11  *u     jj        u     j     «  />.    r tne  "rst  anc> 

"       of  all  the  odd  numbered      "       .     .  =  C.   I  . 

mi  T-  A  /»      T  T-X       last  irom  (j. ) 

The  common  distance  apart  of  ordinates    .     .  =  D.   I 


Then  the  rule  is, 
A  +  4B  +  2C 


X  D  (or  AC)  =  Area,  ABOK    .     .    .  (IX.) 


And  if  more  convenient  (as  it  may  be\  we  transform  this  into  its 
equivalent, 

A  +  4B  +  2C 


D  (or  AE)  _  AreRj  ABOK    m    m     (X  j 

n  applying  this  formula,  it  is  desirable  to  draw  a  figure,  and  num- 
ber all  the  ordinates  (as  below),  commencing  with  1. 

*  Morin's  Mechanics  (Bennett's  Trans.,  I860).—  See  also  Gregory,  Math.  Prac.  Men. 
(1825). 


42 


MEASUREMENT  OF  EARTHWORKS. 


"  The  same  theorem  will  also  obtain,  for  the  contents  of  all  solids, 
by  using  the  sections  perpendicular  to  the  axe,  instead  of  the  ordi- 
nates." 

In  this  form  it  becomes  applicable  to  excavations  and  embankments, 
or  any  similar  solids  relating  to  a  guiding  line,  centre,  or  base  line, 
to  which  the  cross-sections  representing  ordinates  are  perpendicular. 

See  Fig.  20,  copied  below  from 
Hutton,  page  458. 

Button's  Example  3,  p.  462. 

"  Given  the  length  of  five 
equidistant  ordinates  of  an  area, 
or  sections  of  a  solid,  10,  11,  14, 
16,  16,  and  the  length  of  the 
whole  base,  20." 

Then, 

26  +  108  -f  28  ^ 

X  5  =  2/0. 

"  The  area  or  solidity  required" 


This  formula  of  Simpson  (adopted  by  Hutton)  is  evidently  derived 
from  ike  Prismoidal  Formula,  or  it  may  be,  originated  it,  both  having 
the  same  author,  and  their  precedence  unknown. 

(a.) AYe  will  now  give  an  example  of  Hutton's  Method  of 

Equidistant  Ordinates  (adopted  from  Simpson), — giving  two  stations 
of  a  railroad  cut  (each  100  feet  long,  with  a  road-bed  of  18,  and  side- 


+  24) 


Fig.  a 


Hor:  sea:  £Vcr. 


CHAP.    I.— PRELIM.    PROBS.— ART.  10. 


43 


slopes  U  to  1),  .shown  both  in  profile  and  cross-sections.  (See  Figs. 
21  to  26,  inclusive.) 

The  above  figure  is  a  profile,  or  vertical  section  (of  two  stations), 
upon  the  centre  line  of  a  railroad  cut,  with  a  road-bed  of  18,  and  side- 
slopes  of  1J  to  1.  The  horizontal  scale  (/or  convenience)  being  made 
t  of  the  vertical. 

Firstly :  Computing  each  station  separately,  by  Simpson's  Rule  (II.) 


Stations  1  to  3  =      100  =  h. 

Ills.          WIds. 

30  X    90  =    2700  =  26. 
20  X    60  =    1200  =  2  t. 

Stations  3  to  5  =      100  =  h. 

II  ts.          Wids. 

20  X  60=  1200  =  26. 
10  X  30  =    300  =  2  t. 

50  X  150  =    7500  =*  8  m. 

30  X  90  =  2700  =  8  m. 

-5-  by  12)11400 

-T-  by  12)4200 

Mean  Area   .     .  =      950 

X  by  h  .  =          100 

Mean  Area    .     .  =    350 
X  by  h  .  =        100 

Solidity  in  c.  ft.  =      95000 

Solidity  in  c.  ft.  =    35000 

-4-  27  .     .  =        3519 
Deduct     Grade 
Prism  for  100 

feet     .     .     .  =            200 

-^  27  .     .  =      1296 
Deduct      Grade 
Prism  for  100 
feet   .     .     .     .  =        200 

Solidity  inc.  yds.  =          3319 

Solidity  in  c.  yds.  =      1096 

Then,  3319  +  1096  =  4415  cubic  yards,  whole  solidity  of  cut  from 
1  to  5  inclusive. 

Secondly:    Now   computing    the  same,   in  a  body,   by    Hutton's 
Rule  (X.). 

Data. 
(1350 
L=JjL50 

(1500 


B 


5 
337-5 


(  937- 
J  .  337- 
(.1275 


X  4  =  5100 


600     X  2  =  1200 


We  have, 


1500  +  5100  +  1200 


X  100 


Now,  -4-  by  27 

Deduct  Grade  Prism,  200  X  2  stations. 

Solidity  in  cubic  yards 

(The  same  as  above.) 


C.  feet. 

130,000 

4,815 

400 

4,415 


44 


MEASUREMENT  OF  EARTHWORKS. 

(CROSS    SECTIONS.) 


(b.) The  preceding  example  clearly  shows  that  Hutton's 

method  of  equidistant  ordinates  is  merely  the  Prismoidal  Formula 
extended  to  several  stations,  instead  of  confining  it  to  one. 

There  is  another  mode  of  considering  this  question  where  the  cross- 
sections  are  triangular,  and  the  ground  level  transversely. 

Thus,  in  any  station,  let  h  and  h'  be  the  end  hights  from  the  inter- 
section of  the  side-slopes  to  the  ground,  then,  7i2  r  and  hf2  r  =  the  cor- 
responding areas  (r  being  the  slope  ratio,  which,  in  the  preceding 
example  =  1£),  then  omitting  r,  a  common  factor,  we  have  in  A2  and 
hn  vertical  lines,  or  ordinates,  representative  of  the  end  areas,  and  in 

( — - — J  of  the  mid-section. 


CHAP.   I.— PRELIM.   PROBS.— ART.   1O.  45 

The  square  roots,  then,  of  the  areas  (however  computed,  and  what- 
ever be  the  ratio  (r)  of  the  side  slopes),  correctly  represent  them; 
since  these  roots  form  the  side  of  an  equivalent  square  (or  half  base 
of  an  equivalent  triangle,  with  1  to  1  side-slopes) — squaring  which, 
obviously  re-produces  the  areas  they  are  the  roots  of. 

Hence,  the  end  areas  being  given  in  any  station,  or  number  of 
stations,  their  square  roots  may  represent  them  in  Hutton's  rule  of 
cubature,  and  any  pair  of  roots  added  together,  and  their  sum  squared, 
gives  4  times  the  mid-section  between  them  ;  which  is  precisely  what  we 
need  in  the  Prismoidal  Formula. 

This  is  evident,  from  Fig. 
27,  where  we  suppose  h  and  k' 
placed  in  a  continuous  line,  then, 


_v_ 


/h  -4-  h'\2      .    .  „  ,,  *+~ 50 

( — - — j  =  \  the  square  of  (h  -^^ 

-f  hf),  or  equivalent  to  the  pro- 
position of  geometry — that  the  square  of  a  whole  line  equals  4  times  the 
square  of  half. 

f  Let  h  =  30,  and  h'  =  20,  then  h  -f  h'  =  50, h  +  h'  =  25 

4-^)'=  (25)2  =  the  mid-sec.  =  625,  and  X  4  =  2500  \ 

(A  +  h')*  =  (50)' =  2500  j 

VWhile  /i2  =  900    =  one  end  area,  and  hn  =  400,  the  other. 

Also, 

(        h2  +    h'2  -f  2  (h  X  h')  *) 

J  =  900  -f  400  -f-       1200  =  2500  I 
(=  (h   +  hj    ,,    .;..    .  =  2500 j 

From  all  which,  we  readily  draw  the  following: 

Rule. — Compute  the  end  areas  at  each  regular  station  (numbered 
upon  a  diagram  on  Hutton's  plan,  by  the  odd  numbers,  1, 
3,  5,  7,  etc.,  marking  also  the  even  numbers  intermediately, 
which  are,  in  fact,  half  stations,  or  the  places  of  mid-sec- 
tions),— find  the  square  roots  of  these  end  areas: — add  any 
two  adjacent  roots,  and  their  sum  squared  equals  4  times  the 
area  of  the  mid-section,  between  the  regular  stations. 


MEASUREMENT   OF  EARTHWORKS. 


Let  Fig.  28  be  the  profile  of  one  station  of  cutting,  from  intersection 

of  slope  to  ground. 
h  and  h'  =  The  end  hights,  or  representative  square  roots  of 

the  areas,  at  regular  stations,  numbered  odd. 
m  —  The  place  of  the  mid-section,  numbered  even,  and  repre- 
sented by  its  ordinate. 
Length  =  usually,  100,  between  principal  stations. 


.  28. 


Whence, 
h*  +  h'2  +  4m? 


Length. 


6 


X  100  =  Solidity,  by  the  Prismoidal  Formula. 


XI. 


Which,  for  one  station,  is  equivalent  to  Hutton's  Ride. 

(C.)  ......  So  that  having  the  end  areas  given,  we  deduce  at  once 

the  mid-section,  by  a  table  of  roots  and  squares,*  and  can  proceed 
station  by  station,  prismoidally ,  to  find  the  solidity. — Or  combining 
them  as  in  Hutton's  Rule  for  cubature,  we  may  calculate  in  a  body  the 
whole  of  a  cut  or  bank. 

Thus,  taking  the  preceding  example,  and  tabulating  it  (see  Figs. 
21  to  26). 


Stations. 

Areas. 

Even  Nos. 

Odd. 

Even. 

Extreme.  |  Odd  Nos. 

Squares,  or 
Mid-see. 

1 

1350 

36-7423 

Areas. 

2 

61-24 

3750 

3 

600 

24-4949 

4 

36-74 

1350 

5 

150 

12-2475 

1500 

600 

5100 

2 

1200 

A. 

20. 

4  B. 

This  tabulation  may  be  made  in  any  more  convenient  form,  or  the 
data  may  be  written  upon  the  working  profile  of  the  line  with  advantage. 

*  Such  as  Barlow's  (Prof.  De  Morgan's  Ed.,  London,  1860),  which  is  the  most  con- 
venient and  extensive, — or  any  like  tables. 


.  CHAP.  I.— PRELIM.  PROBS.— ART.  10.  47 

Then, 

A      -f-     4B     -f    20       Mean  Area.     Length  of  Sta.        Cub.  Ft. 

1500  +  5100  +  1200  __  1300  X   100  =  130000  =  by  Hutton's 
6  Rule  X. 

'Now,  dividing  by  27, =       4815 

Deduct  grade  prism  for  two  stations  .     .  =         400 

Leaves  solidity  in  cubic  yards  (as  before)  =  4415.  From  1  to  5 
=  200  feet. 

The  division  by  6  in  the  first  term  results  in  a  mean  area,  which  X 
by  length,  gives  the  solidity — and  enables  us  to  use  a  table  of  cubic 
yards  to  mean  areas,  as  soon  as  we  have  found  the  latter,  in  order  to 
obtain  the  cubic  yards  more  readily  by  inspection. 

(d.) In  further  illustration  of  this  important  method  of 

computation  in  earthworks, — we  will  submit  another  example,  repre- 
senting an  entire  railroad  cut,  with  20  feet  road-bed,  and  side-slopes 
of  1  to  1,  laid  off  in  regular  stations  of  100  feet,  and  truncated  at 
both  ends  in  light  cutting  (at  selected  stations),  so  as  to  secure  full 
cross-sections  throughout;  and  also  an  even  number  of  equal  distances 
(apart  sections),  each  100  feet,  or  regular  and  uniform  stations,  what- 
ever their  length. 

These  truncations  are  made  before  proceeding  to  the  calculation, 
so  that  all  the  cross-sections  shall  be  complete  (or  have  some  side 
slope — however  small — at  both  edges  of  the  road-bed),  which  simplifies 
the  main  calculation,  while  in  the  end  the  truncated  volumes  may  be 
computed  independently,  and  added  in  with  the  rest. 

Again,  if  the  ground  should  have  required  the  insertion  of  interme- 
diates in  any  one  or  more  of  the  regular  stations,  it  will  be  best  to 
draw  a  pencil  line  around  all  such  whole  stations  upon  the  diagram, 
and  compute  them  separately  from  the  main  body — the  places  of  such 
stations  being  considered  vacant  for  the  time  (omitting  distance,  mid- 
section,  and  end  areas,  so  far  as  they  apply  to  the  assumed  vacancy), 
and  thus  the  cut  will  be  computable  under  our  rule,  in  one  or  more 
masses  (as  though  a  single  mass  originally),  according  to  the  number 
of  vacant  spaces.  A  little  practice  will  familiarize  this  matter  better 
than  further  explanation,  as  the  object  to  be  attained  is  evident. 


48 


MEASUREMENT   OF  EARTHWORKS. 


Generally,  we  may  compute  the  cut,  or  bank,  in  one  principal 
mass,  and  then  calculate  separately,  and  add. 

1.  The  solidity  in  the  special  stations  containing  intermediates. 

2.  The  quantities  of  work  of  the  same  kind,  at  the  passages  from 

excavation  to  embankment,  at  both  ends  of  the  cut  (as  will 
be  further  explained). 

In  all  such  cases  (indeed,  in  all  cases  of  heavy  work),  it  is  necessary 
to  draw  diagrams,  as  below,  and  these  (in  cross-sections)  will  usually 
have  a  scale  of  20  feet  to  the  inch,  which  long  practice  has  shown  to 
be  entirely  suitable ;  but  any  preferred  scale  may  be  employed,  or  the 
cross-section  paper  in  common  use  amongst  engineers — which  carries 
its  own  scale — and  which  will  be  found  convenient  in  many  respects, 
either  bound  up  for  the  purpose,  or  in  loose  sheets,  to  be  ultimately 
tacked  together,  including  a  mile  forward,  or  thereabouts. 


Profile  of  8  stations  of  railroad  cut;  base  20,  side-slopes  1  to  1. 

a  b  =  Intersection  of  side-slopes,  or  edge  of  diedral  angle,  formed 

by  their  planes  meeting. 

c  d  —  Grade,  or  formation  line  of  the  road-bed  =  +  O'O. 
ef  =  Surface  line  of  ground,  as  cut  by  centre  plane. 
gp  =  Grade  prism — deductive  for  solidity 


Tig.  29                                                                / 

V? 

^^^-  *  5O 

Hor:  b  c  a:  tb  VBT.                                        415  ^f^ 

^s 

k 

* 

LS,^^ 

^^ 

X        Erojile 

41OXX* 

^ 

X+1D 

41 

J—  \ 

r 

S 

a 

8 

> 

X^ 

45^^-«^**n 

a 

«  

C3 

s 

If 

n*° 

±sT 

C 
£    50 

so 

S 

R 

50 

so   p 

a 

,-10 

_m 

h 

1.         «S)        3         (4)        5         (6) 

7         43)         »         00)        U        02)       13         Hi)        15        OC)        17 

Regular  stations  designated  by  odd  numbers  (1,  3,  5,  etc.). 
Mid-section  places  by  even  numbers  (2,  4,  6,  etc.) 


f 

CHAP.   I.— PRELIM.    PROBS.— ART.    10. 


49 


The  ordinates  show  the  level  hlghts  from  grade  to  ground,  to  which 
add  always  the  common  hight  of  grade  triangle. 

Transverse  slopes  are  shown  on  cross-sections. 


f  Regular  Stations      =     !• 
Cross-sectirm  Areas  =  232*5 
I   Square  Boott  =    15  25      1 

1    Sums  of  Roots  =          33-94 


3-  5-  7'  9'  11'          13-          15-          17- 

349-2      412-7      720-5      844-8      1085-        901-5      516-        259-5 
18-69      20-31      26-84      29-06        32-94      30-02      22-72      16-09 
39-00      47-15      55-90      62-00      6296      52'74      38-81 


Squares  of  Sums       =       1151-9    1521-0    2223-1    3124-8    3844-0    3964*0    2781-5    1506-2 
^  These  squares  are  each  equal  to  4  times  the  mid-section,  between  regular  stations. 

All  hights  and  areas  taken  to  intersection  of  slopes. 


Mean  areas  computed  separately 

General  Mean  Area  computed 

for  each  regular  station,  by  Simp- 

by Button's  Rule, 

son's  Rule. 

A  -f  4B  -f  2C 

232-5  ^ 

6 

(1  to  3)         349-2 

1151-9 

• 

Tubulated  fbr  the  numerator  by 

6)1733-6 

successive  additions  —  equivalent  to 

multiplication. 

Mean  Area  =       288'9  , 

349-2  > 

1     .     .         232-5 

2                 1151-9 

(3  to  5)         412-7 

^         •         •               1    1  -  '  I     • 

o           f      349-2 

1521-0 

3    '     '{      349-2 

6)2282-9 

4    .     .       1521-0 

Mean  Area  =       380*5  , 

r                f       412-7 

O          .          .   -\              t  -4  n   fv 

1  to  9 

1      412-7 

412-7  1 

6    .    .      2223-1 

(5  to  7)         720-5 
2223-1 

7            f      720-5 
'     '     '(      720-5 
8     .     .       3124-8 

6)3356-3 

9     .     .        844-8 

Mean  Area  =      559'4 

6)  12062-9 

720-5  ' 

1  to  9  =       2010-5   Gen.  Mean 

(7  to  9)         844-8 

Area. 

3124-8 

. 

Separate  Mean  Areas. 

6)4690-1 

(    288-9 

Mean  Area  =       781*7 

lto'9               J     380'5 

. 

'     '    *  1     559-4 

I    781-7 

Same  as  above  =      2010*5 

50 


MEASUREMENT  OF  EARTHWORKS. 


CHAP.   I.— PRELIM.    PROBS.— ART.   10. 


51 


Mean  areas  computed  separately 
for  each  regular  station,  by  Simp- 
son's Kule. 


(9  to  11) 

Mean  Area  = 
(11  to  13) 

Mean  Area  = 
(13  to  15) 

Mean  Area  = 
(15  to  17) 

Mean  Area  = 

844-8  ' 
1085-0 
3844-0 

• 

6)5773-8 

962-3  J 

1085-0  ' 
901-5 
3964-0 

6)5950-5 

991-8  j 

901-5  1 
516-0 
2781-5 

6)4199-0 

699-8  , 

516-0  > 
259-5 
1506-2 

6)2281-7 

380-3  , 

General  Mean  Area  computed 
by  Button's  Kule. 

A  +  4B  +  2C 
6 

Tabulated  for  the  numerator  by 
successive  additions — equivalent  to 
multiplication. 

Bro't  over  1  tO  9  =  12062-9 

9    .     .        844-8 
3844-0 
1085-0 
1085-0 
3964-0 
901-5 
901-5 
2781-5 
516-0 
516-0 
1506-2 
259-5 


1  to  17 


11 


13 


15 


17 


Gen.  Mean  Area 


Separate  Mean  Areas. 


6)  30267-9 
5044-7 


1  to  17 


Brought  over  =  2010'5 
962-3 
991-8 
699-8 
380-3 


\     Total    .     .  = 
(Same  as  above.) 

Then,    Mean  Area. 

5044-7  X  100  _ 

27  "" 


5044-7 


C.  yards. 

18684-1 


=    2963-2 


Deduct  prade  Prism 
for  8  stations  = 
370-4  X  8  ...  

Solidity =  15721- 

in  cubic  yards  from 
1  to  17. 

So  that  the  final  solidity  of  this  cut  (as  shown)  from  grade  to 
ground,  vertically,  and  from  1  to  17  (8  stations),  horizontally  =  15721 
cubic  yards  (excluding  for  the  present  the  grade  passages). — A  com- 


52 


MEASUREMENT  OF  EARTHWORKS. 


•F5|.35 


cot 


00' 


Fig.  38 


Cross  -sections 


/^n>    l/l> 

CHAP.  I.— PRELIM.   PROBS.— ART.  1O.  ^         ^^63/  > 

^XA 

parison  of  the  calculated  work,  by  Separate  Mean  Areas,  and  by 
General  Mean  Area, — while  resulting  alike,  evinces  the  superiority  of 
the  latter,  in  point  of  brevity. 

In  the  tabulation  for  General  Mean  Area,  it  will  be  observed  that 
the  extreme  end  areas  are  written  but  once  (equivalent  to  addition) 
— the  odd  numbered  areas  twice  (equivalent  to  X  by  2),  while  the 
even  numbered  areas  are  written,  in  effect,  4  times, — as  squares  of 
sums  of  adjacent  representative  hights,  because  in  that  shape  they  each 
equal  4  times  the  area  of  the  prismoidal  mid-section. 

(6.) We  must  now  consider  the  passages  from  excavation 

to  embankment  at  both  extremities  of  the  cut,  near  the  regular  sta- 
tions, 1  and  17,  where  it  was  assumed  to  be  truncated,  in  order  to  sim- 
plify its  computation. 

Figs.  39  to  42  show  these  passages  so  clearly,  in  the  assumed  case, 
as  to  need  little  explanation. 

On  plain  ground  the  line  of  passage  a  c  will  often  be  so  nearly 
normal  to  the  centre  that,  having  set  the  grade  peg  in  the  centre  line 
at  e  (the  entrance  of  the  cut),  we  may  place  those  for  the  edges  ofthe 
road-bed  (as  a  and  c),  at  right  angles  in  many  cases,  where  the  ground 
differs  in  level  only  a  few  tenths  of  a  foot;  the  error  being  merely  a 
change  of  some  yards  from  excavation  to  embankment,  which  is  quite 
immaterial,  since  their  values  differ  little  per  cubic  yard. 

But  where  the  ground  is  much  inclined,  in  either  direction,  the 
grade  pegs  aec  must  be  set  on  an  oblique  line,  broken  at  e,  if  neces- 
sary. 

Precise  rules  can  scarcely  be  furnished  for  such  cases,  but  the 
quantities  being  usually  small,  and  the  distances  short,  any  of  the 
ordinary  methods  may  be  safely  employed. 

In  the  case  before  us,  we  have  made  the  computation  from  17  to  a, 
and  from  1  to  a,  by  the  Arithmetical  Mean,  and  for  the  parts  from 
a  to  c  as  pyramids. 

In  this  manner  we  have  found  the  volume  of  excavation,  at  the 
passage  at  Fig.  39,  to  be =  321  cubic  yards. 

And  at  Fig.  41 =  622     " 

Total,  in   the  whole  length   of   the  passages       

(230  feet) =  943  cubic  yards. 


54 


MEASUREMENT   OF  EARTHWORKS. 


So  that,  finally,  we  have  for  the  solidity  of  the  entire  railroad  cut, 
under  consideration,  the  following  result : 

From  1  to  17  (as  before  computed)  =  15721  cubic  yards. 
In  the  passages  from  excavation  to 

embankment,  at  both  ends  (230 

feet  long  in  all) =      943      "        " 

Whole  solidity  of  the  cut  from  grade 
to  grade,  on  both  sides    .    .     .  =  16664  cubic  yards. 

We  will  now  illustrate  the  passages  from  excavation  to  embank- 
ment, at  both  ends  of  the  cut  (shown  in  profile  at  Fig.  29.) 


In  Figs.  39  to  42  all  letters  refer  to  similar  parts. 
1  and  17  =  Places  of  cross-sections,  at  the  selected  regular  stations, 

where  the  cut  was  truncated,  to  obtain  full  work. 
a  a  =  Cross-section,  where  one  edge  of  road-bed  runs  to  grade. 

c  =  Grade  point  at  the  other  edge,  or  opposite  side. 
a  c  =  Line  of  junction  of  cut  and  bank,  at  grade  level. 
bb  —  Slopes  of  cut. 
d  d  =  Slopes  of  bank. 
e  =  Grade  point  at  centre. 


Total  length  of  cut  between  the  extreme  grade  points  forming  the 
vertices  of  the  small  pyramids  at  c  and  c  =  1030  feet. 


CHAP.   I.— PRELIM.    PROBS.— ART.   11. 


55 


Other  modes  may  be  used  for  treating  the  question  of  passages 
between  excavation  and  embankment,  but  the  above  is  as  simple  as 
any,  and  may  be  easily  modified  for  particular  cases. 


11.  With  Railroad  Cross-sections  in  Diedral  Angles — to  find  the  mid- 
section  of  the  Prismoidal  Formula,  by  a  brief  calculation  from  the  End 
Areas,  without  a  Special  Diagram. 

In  all  railroad  cross-sections,  instrumental  data  of  adequate  extent 
are  first  obtained  in  the  field  by  well-known  processes,  and  these  data 
enable  us  in  the  office,  subsequently,  to  draw  them  as  diagrams,  by  a 
suitable  scale,  and  to  compute  their  superficies. 

The  length  of  each  separate  solid  of  earthwork,  and  its  position 
upon  the  centre  or  guiding  line,  is  also  known. 

With  these  given  data,  the  Prismoidal  Formula  requires  the  deduc- 
tion of  a  hypothetical  mid-section,  in  some  form,  for  use  under  the 
general  rule,  or  its  modifications. 

As  mentioned  previously,  this  mid-section  is  usually  derived  from 
the  Arithmetical  Average  of  like  parts  in  the  end  sections,  and  even 
in  extremely  irregular  ground,  to  find  this  leading  section  of  an  Earth- 
work Prismoid,  is  not  very  difficult — when  the  diagrams  of  the  end 
cross-sections  are  correctly  drawn — (as  in  heavy  work  they  always 
should  be),  or  even  from  the  field  notes  of  the  engineer,  since  the  posi- 
tion of  every  leading  point  of  ground,  transversely,  is  always  fixed 
and  recorded  by  level  bights,  and  distances  out  from  centre,  and  their 
average  position  is  always  reproduced,  proportionally,  -in  the  mid- 
section. 

Nevertheless,  some  judgment  is  required  in  deducing  the  mid-sec- 
tions from  the  end  ones,  by  Arithmetical  Means,  since  the  points  to 


56  MEASUREMENT  OF  EARTHWORKS. 

average  upon  are  often  in  doubt, — the  process,  too,  including  finding 
its  area,  is  like  most  others  connected  with  earthwork  computations, 
very  often  tedious,  so  that  some  shrewd  mathematicians,  while  con- 
ceding the  accuracy  of  this  method,  when  properly  carried  out,  have, 
nevertheless,  deemed  it  unsatisfactory  in  some  respects.* 

It  is  well,  therefore,  to  have  the  means  of  operating  with  given  end 
areas,  to  find  the  mid-section,  without  the  necessity  of  arithmetically 
deducing,  or  even  of  sketching  it. 

We,  therefore,  now  submit  some  rules  and  examples  by  which  the 
area  of  the  mid-section  may  be  computed  from  the  ends,  without 
deriving  it  in  the  usual  way,  or  drawing  for  it  a  special  diagram. 

These  rules  are  intended  only  for  Earthwork  Prismoids,  within  die- 
dral  angles ;  and  though  their  range  is  clearly  more  extensive,  the 
variety  of  prismoidal  solids  is  so  great  that  it  is  probably  best  to  limit 
our  rules  and  examples  to  the  object  before  us. 

The  broken  ground  line  of  very  irregular  cross-sections  should 
always  be  reduced  to  a  uniform  slope,  by  a  single  equalizing  line  (or 
at  most  by  two),  containing  exactly  the  same  superficies,  by  the  method 
of  Art.  8, — and  the  bights  and  widths  ascertained  for  each  section 
(by  the  equalizing  line),  and  verified  by  multiplication  to  re-produce 
the  area  equalized, — see  8  (a), — these  bights  and  widths  enable  us  at 
once  to  compute  the  volume  of  the  prismoid  by  Simpson's  Rule  (their 
product  giving  end  areas) — (Art.  2  (a)  ) — and  the  sums  of  these 
bights  and  widths,  when  multiplied  together,  producing  always  8 
times  the  mid-section  (without  directly  deducing  it). 

Having  given  then  the  end  areas,  or  the  bights  and  widths  which 
produce  them,  we  readily  find  the  Prismoidal  Mid-section  by  the 
following : 

x  Arithmetical  Mean  -f  Geometrical  Mean 

(1.)  —  — •= —  .  =  Mid-sec. 

2 

(Sum  of  square  roots  of  end  areas)2  _        . 

tSum  end  bights  X   sum  end  widths  ,_, , 

(3.) .     .  —  Mia-sec. 

(4.)  By  the  method  of  Initial  Prismoids— Art.  3  (a). 

I    *  Warner's  Earthwork  (1861).— Davies'  New  Surveying  (1870). 

f  These  bights  and  widths  (used  in  3)  are  those  connected  with  the  equalizing  line 
of  the  equivalent  triangular  section — the  product  of  which,  at  each  cross-section,  re-pro- 
duces exactly  the  double  area  of  the  whole  surface,  from  the  side-slopes  to  the  broken 
ground  line;  and  the  product  of  their  sums  always  equals  eight  times  the  mid-section. 


Rules. 


CHAP.   L—  PRELIM.   PROBS.— ART.   11. 


57 


Other  rules  might  be  given,  but  these  Jour  appear  to  be  the  simplest 
and  best  for  use  in  earthwork,  under  the  view  we  have  herein  taken. 

Having  then  found  the  mid-section,  and  having  the  end  areas  and 
length  previously  given,  we  can  easily  compute  the  volume  of  any 
earthwork  solid,  by  the  Prismoidal  Formula,  or  its  numerous  modifi- 
cations. 

1.  A  Prism  .  =      Base. 


By  Geometry,  we   have 
for  the  mid-sections  of 


f  A  Wedge,  with  back  ^ 

2.  <       and  edge  equal  and  >    =  i  Base, 
v.      parallel    .     .     .     .  J 

3.  A  Pyramid =  1  Base. 

Fig.  43  shows  the  end  cross-sections  of  one  station  of  a  railroad  cut, 
upon  irregular  ground,  both  upon  one  diagram*,  road-bed  20,  side- 
slopes  1  to  1.  Length  of  station,  100  feet. 


g  g'g"  cent:  of  grav: 


Centre  higbts  to  intersection  of  slopes. 

-f  37-5 

4-  31-6 

+  25-7 

from  equalizing  line. 


Total  widths  from  side  to  side. 

80 
68 
56 


58 


MEASUREMENT  OF  EARTHWORKS. 


Note: 

Both  in  Figa.  43  and  44  the  same  letters  refer  to  like  parts. 
CC  =  Centre  line  of  railroad,  or  guiding  line  of  earthwork, 
a  b  sm  Equalizing  line  of  broken  ground  surface  of  larger  end     . 
«/  =         "  "  "  "  "         of  smaller  end . 

« d  =         "  "  "  "  "          of  mid-section  . 


14°    2'  slope. 
,  15°  57'     " 
14°  50'     " 


Fig.  44,  like  the  preceding,  shows  both  end  sections  of  a  railroad  cut, 
upon  one  diagram.  Road-bed  =  20,  side-slopes  1  i  to  1.  Length  =  100. 


Centre  bights  to  intersection  of  slopes. 

+  22*02 
+  26-07 
+  29-81 
from  equalizing  line. 


Total  widths  from  side  to  side. 

66- 

78-7 

90-7 


In  this  figure  (44)  the  line  ef  has  a  minus  slope,  which  is  always 
the  case  when  the  area  assumed  up  to  .the  equalizing  point  is  greater 
than  that  to  be  equalized. 

In  both  of  the  above  figures,  I  is  the  intersection  of  the  side-slopes, 
or  edge  of  the  diedral  angle,  containing  the  earthwork  prismoids. 

The  constant  area  of  the  grade  triangle,  with  side-slopes  of  1  to  1 
(Fig.  43)  =  100.  While,  with  side-slopes  of  11  to  1  (Fig.  44)  = 
66f.  The  road-bed,  or  graded  width,  in  both  cases  being  20  feet. 
The  altitude  of  this  triangle  for  1  to  1  =  10,  and  for  IHo  1  =  6f. 


CHAP.   I.— PRELIM.   PROBS.— ART.  12.  59 

The  rules  (numbered)  above,  for  the  figures  shown,  give  the  follow- 
ing results : 

(  Fig.  43  gives  Mid-sections  (1)  =  1074-5;  (2)  =  1074-5   ;  (3)  =  1074-4   ;  (4)  =  1074-6 
\  Fig.  44  gives  Mid-sections  (1)  =  1015'   ;  (2)  =  1014-74  j  (3)  =  1015-22;  (4)  =  1015- 

The  small  variations  arise  from  the  decimals  not  being  sufficiently 
extended. 


12.  To  find  the  Prismoidal  Mean  Area  from  the  Arithmetical  or 
Geometrical  Means,  or  the  Mid-section,  by  Corrective  Fractions  of  the 
Square  of  the  Difference  of  End  Hights. 

In  all  cases  we  suppose  the  end  areas  of  the  Prismoid  to  be  given, 
and  that  the  Prismoid  itself  is  contained  within  a  diedral  angle,  the 
plane  angle  measuring  it  being  supplemental  to  double  the  angle  of 
side-slope,  as  in  the  Figs.  43  and  44. 

The  simplest,  and  probably  by  far  the  most  generally  employed 
method  of  finding  a  mean  area  between  two  others, — is  by  the  Arith 
metical  Mean — which  is  itself  half  the  sum  of  any  two  magnitudes. 

Adopting  the  Arithmetical  Mean  as  being  the  simplest  known 
base,  and  forming  all  sections  of  earthwork  by  prolonging  the  planes 
of  the  side-slopes  to  their  intersection  (or  supposing  them  to  be),  so 
as  to  bring  the  computed  prismoids  within  diedral  angles  of  given 
divergency. 

We  have,  from  the  relations  between  the  sums  or  differences 
of  the  squares,  or  rectangles  of  lines  producing  areas,  some  rules, 
which  may  often  be  useful  in  the  calculation  of  earthwork,  for  cor 
recting  mean  areas  to  be  used  in  finding  the  solidity. 

This  correction  being  always  equivalent  to  some  fraction  of  the  square 
of  the  difference  of  the  end  hights. 

While  these  end  hights  are  always  to  be  deemed  and  taken  as  the  squart 
roots  of  the  end  areas,  and  are,  in  fact  (as  before  mentioned),  a  side  of 
an  equivalent  square,  or  half  base  of  an  equivalent  triangle,  having 
side-slopes  of  1  to  1  (or  a  diedral  angle  of  90°), — for  (we  repeat),  no 
matter  what  may  be  the  ratio  of  actual  side-slope,  nor  how  irregular 
the  ground  surface,  the  square  root  of  the  area  is  invariably  the  true 
representative  hight  whichx  rectifies  the  section,  and  which,  when 
squared,  reproduces  the  area. 

See  Art  10  (a)  (b)  etc.,  where  much  use  is  made  of  these  square 
roots,  or  representative  hights. 


60 


MEASUREMENT  OF  EARTHWORKS. 


Having,  then,  the  end  areas  given,  and  their  square  roots  or  hights 
ascertained, 

D   =  Difference  of  hights. 

D2  =  The  square  of  the  difference  of  hights. 


Rules: 


(1)  Arithmetical  Mean 


Sum  end  areas 


(2) 
(3) 
(4) 


(5) 


(6) 


Then  the  Prismoidal  Mean  Area. 
.     .  =  Arithmetical  Mean  —  I  D2. 
.     .  =  Mid-section   .     .     .  -f  ^  D2. 
.     .  s=s  Geometrical   Mean  +   J  D2. 

Prismoidal  Mid-section. 
.     .  =  Arithmetical  Mean  —  J  D2. 

Geometrical  Mean. 
.     .  =  Arithmetical  Mean  —  \  D2. 


^  For  Fig.  43  these  rules  give,  For  Fig.  44  these  rules  give, 

(1)=»1039-       =Arith.  Mean. 

(2)  =  1022-9  ) 

(3)  =  1023-    V=  Pris.  Mean. 

(4)  =  1023-2  ) 

(5)  =  1014-8     =  Pris.  Mid-sec. 
(6)=    991-       =Geom.  Mean. 

In  these  numerical  illustrations  (as  in  others)  slight  variations 
arise  from  insufficient  decimals. 

Baker*  gives  yet  another  rule  for  the  Prismoidal  Mean  Areas,  as 
follows : 


(1)  =  1110- 

=  Arith.  Mean. 

(2)  =  1086-4 

i 

(3)  =  1086-3 

»•  =  Pris.  Mean. 

.(4)  =  1086-4 

\ 

(5)  =  1074-6 

=  Pris.  Mid-sec. 

(6)  =  1039-2 

==Geom.  Mean. 

Sum  end  areas  -f  Rectangle  hights 


Prismoidal  Mean. 


And  we  may  repeat,  as  another  modification  of  the  Prismoidal 
Formula,  arising  from  this  discussion,  the  following  (same  as  XI., 
before  given) : 

XII Solidity 

_  (Sum  of  squares  of  hights)  -f  (Square  of  sum  of  hights) 
—  _  .  y^  fa 


*  Baker's  Railway  Engineering  and  Earthwork  (London,  1848).  Other  writers  have 
given  the  same,  and  it  is  deducible  from  Button's  Mens.,  Prob.  7,  as  most  of  these  For- 
mulcts  are. 


CHAP.  I.— PRELIM.   PROBS.— ART.  12. 


61 


2  (Sum  sqs.)  +  2  (Rect.  hights) 


,  or  -f-  2  = 


This  is  equivalent  to 

(Sum  of  sqs.)  -f-  (Rect.  hights) 

— ,  which  is  Baker  s  rule  above,  or  Bid- 
der's, as  quoted  by  Dempsey  (Practical  Railway  Engineering  (4th 
edition)  1855). 

We  may  illustrate  this  matter  further  by  two  simple  figures. 


Here  Fig.  45  represents  a  1  to  1  side-slope — diedral  angle  90°  ;  and 
Fig.  46  a  side-slope  of  H  to  1— diedral  angle  112°  38'. 

In  both  these  diagrams  the  same  letters  refer  to  like  parts. 


/    ..L«J 


References. 
CO  =  Centre  line. 

I     =  Intersection  of  planes  of  side-slope. 
a  b  =  Ground  line  of  one  end  section. 
c  d  =       "      "    %  of  the  other. 
m  s  —       "      "       of  the  mid-section. 
Hights  and  areas  both  extend  to  the  intersection  at  I. 


62  MEASUREMENT   OF  EARTHWORKS. 

In  Fig.  45,  The  end  areas  are  1600  and  400— the  higlits  40  and 
20— and  by  the  rules  herein,  Arithmetical  Mean  = 
1000,  Geometrical  Mean  =  800,  Mid-section  =  900, 
Prismoidal  Mean  Area  =  933£,  by  all  the  rules. 

In  Fig.  46,  The  end  areas  are  2400  and  -600— the  hights  =  48'99 
and  24'99,  being  the  square  roots  of  the  respective 
end  areas — and  by  the  rules  herein,  Arithmetical 
Mean  =  1500,  Geometrical  Mean  =  1200,  Mid-sec- 
tion =  1350,  Prismoidal  Mean  Area  1400,  by  all  thg 
rules. 

The  areas  and  hights,  in  both  examples,  are  contained  between  the 
ground  lines,  and  the  intersection  of  the  planes  of  side-slope,  or  edge 
of  diedral  angle,  including  the  Prismoid  of  Earthwork. 

13.  Applicability  of  the  Prismoidal  Formula  to  find  the  Solidity  of 
Various  Solids  other  than  Prismoids. 

The  Prismoidal  Formula  appears  to  be  the  fundamental  rule  for  the 
mensuration  of  all  right-lined  solids,  and  the  special  rules  given,  in 
works  on  mensuration,  for  ascertaining  the  volume  of  solids  in  general 
use,  seem  like  mere  cases  of  the  former ;  though  their  relation  has  never 
been  demonstrated  in  plain  terms  by  mathematicians — so  as  to  con- 
nect them  directly — further  than  prisms,  pyramids,  and  wedges,  which 
has  already  been  done  by  the  present  writer  in  Jour.  Frank.  Inst., 
1840. 

Nevertheless,  Hutton  (1770)  has  indicated  numerous  applications, 
and  various  writers  have  since  shown  the  applicability  of  the  Pris- 
moidal Formula  to  ordinary  solids,  and  also  its  coincidence  with  many 
special  rules  of  the  books,  when  proper  algebraic  substitutions  are 
made;  and  it  has  been  further  shown  to  hold  for  certain  warped 
solids,  to  which  its  application  was  not  expected.* 

As  an  evidence  of  its  remarkable  flexibility,  we  may  show,  briefly, 
its  application  to  the  three  round  bodies,  illustrated  by  a  diagram. 

(1)  The  volume  of  a  cone  equals  the  product  of  its  base  X  i  its  hight.'f 
The  prismoidal  mid-section  of  a  cone  =  \  the  area  of  the  base.  The 
section  at  the  top,  or  vertex  =  0.  Then,  the  sum  of  these  areas  used 
prismoidally  =  2  base,  which,  X  i  h  =  base  X  i  hight,  which  is 
the  geometrical  rule. 

*  Gillespie,  Frank.  Inst.  Jour.  (1857  and  1859).— Warner's  Earthwork  (1861). 
f  Chauvenet,  ix.  3,  7, 14,  Geom.  (1871).— Borden's  Useful  Formulas  (1851).— Henck'a 
Field  Book  (1854),  Art.  112. 


CHAP.  I.— PRELIM.   PROBS.— ART.  13. 


63 


(2)  The  volume  of  a  sphere  equals  4  great  circles  X  i  fa  radius.* 
Now,  the  prismoidal  sections  at  the  poles  are  both  =  0.     While  four 
times  the  mid-section  =  4  great  circles.     Then,  the  prismoidal  sum 
of  areas  =  4  great  circles,  which  X  i  hight,  or  diameter,  or  £  radius, 
is  the  geometrical  rule. 

(3)  The  volume  of  a  cylinder  equals  the  product  of  its  base  by  its 
hight.*     Now,  by  the  Prismoidal  Formula,  base  -f-   top  +  4  times 
mid-section  =  6  base  (for  all  the  sections  are  alike),  and  6  base  X  ^ 
h  =  base  X  hight,  which  is  the  geometrical  rule. 

So  that  there  can  be  no  doubt  of  the  applicability  of  the  Prismoidal 
Formula  to  the  three  round  bodies;  and  in  a  similar  manner  it  is  easy 
to  show  its  coincidence  with  many  special  rules  for  solids,  but  a  direct 
mathematical  demonstration  connecting  all  these  together,  and  exhib- 
iting their  geometrical  relations,  has  never  come  under  the  writer's 
notice  ;  though  indirectly,  and  perhaps  quite  as  satisfactorily,  this  con- 
nection  has  been  clearly  established  for  all  the  leading  solids  in  prac" 
tical  use. 

Numerical  calculation  of  the  three  round  bodies,  supposing  each  to 
have  a  diameter  of  1,  and  an  altitude  of  1. 


CONE. 

SPHERE. 

CYLINDER. 

Prisnioidfilly. 

Geom.  Rule. 

Prismoidally. 

Geom.  Rule. 

Prismoiditlly. 

Geom.  Rule. 

Top  .    .  =  -0    . 
Mid.X  4  =   '7854 
Base.     .=    7854 
6)1-6708 

Base      =  -7854 

ix^ 

Solidity  =  "2618 

Top  .    ..^.-O- 
Mid.X  4  =3-1416 
Base.     .  =0- 

4  great  circles 
=    3-1416 

XlA°t*A 

Top.    .=    -7854 

Mid.X  4  "=  3'1416 
Base.     .=    -7854 

Base    .  =  -7854 
1 

6)3-i41G 

Solidity**  -6-236 

6)4-7121 

S»hdity=  -7854 

•2618 
1 

Solidity  =   -26T8 

•523«i 
.':'  > 
Solidity  =   -5236 

XI 
Solidity  =    -7854 

c 

a  b  = 

The  Base. 

Flg.W. 

/w\ 

c  d  = 

"    Top. 

IU 

V/     V 

s* 

'  m  s  = 

"    Mid-section. 

h 

1. 

The  common  rules  of  mensuration  are  drawn  from  geometry — but 
geometry  also  teaches  that  a  cone,  a  sphere,  and  a  cylinder,  dimen- 
sioned and  situated  as  shown  by  their  right  sections,  in  Fig.  47,  have 


.»  Chauvenet,  ix.  3,  7,  14,  Geom.  (1871).— Borden's  Useful  Formulas  (1851).— Henok'a 
Field  Book  (1854),  art.  112. 


64  MEASUREMENT  OF  EARTHWORKS. 

their  volumes  in  the  ratio  of  the  numbers  1,  2,  and  3. — Now,  the 
above  calculations  show  the  same  result  numerically,  which,  with  the 
preceding  observations,  furnish  an  adequate  demonstration. 

In  like  manner  we  might  show  that  the  Prismoidal  Formula  applies 
to  all  the  separate  geometrical  solids,  which,  when  aggregated,  form 
the  irregular  prismoid  known  as  an  Earthivork  Solid. 

Now,  considering  this  species  of  solid  as  a  prismoid,  within  the 
limits  of  Button's  definition  (1770),  we  find  that  all  such  admit  of 
decomposition  into  Prisms,  Prismoids,*  Pyramids,  or  Wedges  (complete 
or  truncated),  or  some  combination  of  them,  having  a  common  length, 
or  hight,  equal  to  the  distance  between  the  end  areas  or  cross-sections, 
and  either  separately  or  together  computable  by  the  Prismoidal  Formula 
as  a  general  rule  for  all. 

By  a  similar  analogy  (to  the  three  round  bodies),  we  find  somewhat 
like  relations  to  obtain  between  what  we  may  call  the  three  square  or 
angular  bodies;  which  geometry-shows  to  exist  alike  amongst  them 
all,  the  round  bodies  being  referred  to  the  cylinder;  the  square  or 
angular  ones  to  the  cube. — But  the  wedge  requires  this  special  defini- 
tion, that  the  edge  be  double  the  back. 

1.  A  Pyramid,  with  a  square  base,  on  a  side  of  1,  and 

having  also  an  altitude  of  1,  has  a  volume    .     .     .    ..  =  |. 

2.  A  Wedge,  doubled  on  the  edge,  with  a  square  back,  on  a 

side  of  1,  the  edge  parallel  =  2  (or  double  the  back), 
and  an  altitude  of  1,  has  a  volume =  I. 

3.  A  Cube,  or  Hexaedron,  with  its  six  square  faces,  each 

formed  upon  a  side  of  1,  has  a  volume =  1. 

So  that,  finally,  we  have,  both  in  the  three  round,  and  in  the  three 
square  bodies  (as  defined)  where  unity  is  the  controlling  dimension, 
like  ratios  of  volume. 

Thus,  these  six  bodies, 


(  Cone  and 
\  Pyramid. 


Sphere  and 

Wedge 
(doubled  on  the  edge). 


Cylinder  "]      Solids    of 
and  Cube.  |       Circular 
and 


ra"oTofthvoiuml  =  1-  2-  3»         J  Square  Bases. 

And  of  each  and  all  of  these  alike,  the  Prismoidal  Formula  gives  the 
Solidity, 

*  The  Rectangular  Prismoid  being  always  divisible  into  two  \vedges. 


CHAP.  L— PRELIM.  PROBS.— ART.  14.  65 

14.  Transformation  of  Areas  into  Equivalent  ones,  Simpler  in  Form, 
and  of  Solids  into  Equivalents,  more  readily  Computable  by  the  Pris- 
moidal  Formula,  or  its  Modifications. 

Hutton  hath  defined  a  Prismoid  as  follows: 

"  A  Prismoid  is  a  solid  having  for  its  two  ends  any  dissimilar 
plane  figures  of  the  same  number  of  sides,  and  all  the  sides  of 
the  solid  plane  figures  also."  (Quarto  Mens.,  1770.) 

This  is  the  oldest  and  best  definition  of  the  Prismoid  which  we  are 
able  to  find  on  record.* 

Under  this  definition,  for  which  the  General  Rule  (coinciding  with 
Simpson's)  was  framed  by  Hutton,  it  is  clear  that  we  ought  not  to 
expect  of  the  Prismoidal  Formula  the  cubature  of  curvilinear  solids, 
though,  by  a  happy  coincidence,  it  applies  to  many  such,  which  are 
not  prismoids  at  all,  nor  in  the  least  resemble  them,  geometrically. 

But  though  often  true  of  this  remarkable  formula,  where  a  correct 
mid-section  can  be  first  obtained,  it  by  no  means  follows  that  its 
numerous  modifications  (all  framed  for  right-lined  solids)  will,  like 
their  principal,  also  hold,  as  it  does  in  many  singular  cases  exactly,  and 
in  most  others  approximately. 

It  was  early  discovered  that  it  would  materially  simplify  the  com- 
putation of  irregular  prismoids,  to  transform  them  into  equivalent 
right-lined  bodies,  of  which  the  nature  was  better  known,  and  the 
forms  more  regular  and  simple. 

As  the  calculations  for  level  ground  were  obviously  the  most  easy, 
Sir  John  Macneill,  in  his  Tables  of  1833,  adopted  for  the  end  sections 
the  principle  of  transformation  into  level  hights,  to  contain  equivalent 
level  areas — and  was,  in  fact,  the  originator  of  what  has  since  been 
known  as  the  Method  of  Equivalent  Level  Hights — by  means  of  which, 
the  end  sections  of  irregular  prismoids  of  earthwork  are  transformed 
into  level  trapezoids,  which  are  then  employed  to  compute  an  equiva- 
lent solid  of  the  same  length,  and  transversely  level,  at  top  or  bottom, 
according  as  it  may  be  excavation  or  embankment — each,  however, 
representing  the  other,  when  inverted. 

Sir  John  Macneill  has  been  followed,  more  or  less  closely,  by  most 
of  the  authors  of  Earthwork  tables,  the  bulk  of  which  are  applicable 
to  level  ground  alone,  or  ground  reduced  to  such  ; — though  Watner's 
System  of  Earthwork  Computation  (1861)  deals  with  ground  how- 
ever sloping,  or  even  warped,  within  certain  limits. 

*  See  also  Henck's  Field  Book  (1854).— Davies  Legendre  (1853).— Haswoll's  Mens. 
(1863).— Bonnycastle's  Mens.  (1807).— Hawnev's  Mens.  (1798).  All  define  the  Pris- 
moid  ni  a  right-lined  solid. 

5 


66  MEASUREMENT   OF  EARTHWORKS. 

The  method  of  using  Equivalent  Level  Hights  (when  the  cross- 
section  of  the  ground  is  not  level)  has  been  concisely  explained,  by  a 
recent  writer,  to  consist  in  finding* 

1.  "  The  area  of  a  cross-section  at  each  end  of  the  mass." 

2.  "  The  hight  of  a  section,  level  at  the  top,  equivalent  in  area  to 

each  of  these  end  sections." 

3.  "  From  the  average  of  these  two  hights,  the  middle  area  of 

the  mass." 

"  And,  lastly,  in  applying  the  Prismoidal  Formula  to  find  the 
contents." 

It  is  obviously  necessary  then  to  understand  what  is  meant  by 
equivalency — and  this  we  find  from  Geometry .f 

1 .  "  Equivalent  (plane)  figures  are  those  which  have  the  same 

surface — measured  by  the  area." 

2.  "Equivalent  solids  are  those  which   have  the  same  bulk  or 

magnitude." 

"  Theorem:  If  two  solids  have  equal  bases  and  hights,  and  if 
their  sections  made  by  any  plane  parallel  to  the 
common  plane  of  their  bases  are  equal,  they  are 
equivalent." 

Now,  the  transformation  of  triangular  prismoids  of  earthwork,  by 
means  of  Equivalent  Level  Hights,  meets  every  point  of  Professor 
Peirce's  definitions  of  equivalency,  and  hence  the  solid  they  produce 
may  be  regarded  as  equivalent  to  the  original  defined  by  Hutton : — in 
the  above  theorem,  equality  of  sections  evidently  means  equality  in 
area,  and  not  geometrical  equality,  which  is  somewhat  different. 

Some  writers  have  doubted  the  accuracy  of  the  transformation  or 
equivalency  produced  by  Equivalent  Level  Hights,J  but  it  is  because 
the  solids,  which  they  found  in  error,  were  either  not  prismoids  at  all, 
or  else  the  data  used  were  inadequate  to  the  solution  of  the  problem. 

An  error  in  this  direction  is  not  surprising  ;  for  when  we  know  that 
the  Prismoidal  Formula  applies  correctly  to  a  solid,  we  are  apt  to 
infer  that  its  modifications  also  do, — and  here  the  error  lies. 

For.instance,  we  know  this  formula  does  apply  correctly  to  a  sphere, 
but  if  we  test  that  solid,  by  the  method  of  Equivalent  Level  Hights, 
we  should  find  that  the  end  sections  being  0,  have  a  hight  of  0,  and 
that  the  mid-section  being  constructed  on  a  mean  of  like  parts  in  the 

*  Henck's  Field  Book  (1854).  f  Peirce's  Plane  and  Solid  Geom.  (1837). 

J  Gillespie,  Frank.  Inst.  Jour.  (1859). 


CHAP.   I.— PRELIM.    PROBS.— ART.  14. 


67 


ends  must  also  equal  0,  and  hence  we  might  in  this  way  legitimately 
come  to  the  conclusion  that  the  globe  itself  had  a  solidity  of  0  !  This 
shows  that  Equivalent  Level  Hights  are  limited  in  range. 

The  error  obviously  is — that  all,  or  most  of  the  transformations  and 
modifications  of  the  Prisrnoidal  Formula,  are  intended  for  right-lined 
solids,  "  varying  uniformly  "  from  end  to  end,  like  a  stick  of  timber 
dressed  off  tapering,  and  to  all  such  rectilinear  solids  they  do  apply 
correctly ;  but  not  to  those  which  bulge  out,  or  curve  in,  by  laws 
unknoivn  to  Huttoris  definition  of  the  Prismoid. 

It  would  be  easy  to  illustrate  this  by  examples,  and  to  show  that, 
confined  within  proper  limits,  the  usual  modifications  of  the  Prismoidal 
Formula  are  correct  enough  for  practical  use  ;  but  they  have  not  the 
wide  range  of  their  principal;  nor  must  they  be  expected  to  apply 
either  to  the  three  round  bodies,  or  to  warped  solids,  but  only  to  right- 
lined  ones,  varying  uniformly,  or  nearly  so,  from  end  to  end. 

One  important  point,  however,  must  not  be  overlooked  in  applying 
the  Prismoidal  Formula  (or  its  modifications)  to  cases  of  earthwork: 
that  is,  the  ground  must  be  properly  cross-sectioned;  or,  have  its  sections 
judiciously  located,  while  the  hights  and  distances  of  its  controlling 
points  are  correctly  measured  and  recorded,  prior  to  undertaking  the 
calculations  of  solidity. 

It  is  in  this  point  that  Borden's  ridge  and  holloiv  problem  fails* 
Had  one  or  more  intermediate  cross-sections  been  adopted  there,  no 
difficulty  would  have  existed  in  its  calculation,  either  by  Borden  him- 
self, or  by  subsequent  students. 

To  illustrate  this  subject,  we  will 
give  an  example,  drawn  from  Simp- 
son's original  Prismoid  of  1750,  on 
which  he  founded  the  Prismoidal 
Formula,  or  used  to  explain  it. 
Art.  2,  Fig.  2.  (And  see  Figs.  48, 
49,  50,  51.) 


2.1. 82 


Here  we  will  take  the  Prismoid  as 
being  cut  in  two,  by  the  diagonal 
plane,  through  DB,  so  as  to  divide  it 
into  triangular  prismoids,  and  then 
calculate  one  of  these  halves  in  three 
ways. 


Fig.  49 


*  Borden's  Useful  Formulas,  etc.  (1851).— Henck's  Field  Book  (1854). 


68 


MEASUREMENT  OF  EARTHWORKS. 


1.  By  Simpson's   Rule,  as   the 

half  of  a  rectangular  pris- 
moid,  dimensioned  as  in 
Fig.  2. 

2.  By  Hights  and  Widths,  as  a 

triangular  earthwork  solid, 
with  unequal  side-slopes. 
(See  Figs.  48,  49.) 

3.  By  Equivalent  Level  Hights 

purely  as  an  equivalent  tri- 
angular prismoid,  or  earth- 
work solid,  within  a  diedral 
angle  of  90°,  and  having 
equal  side-slopes  of  1  to  1. 

In  all  these  figures  the  angle  A  =  90°. 
B  and  B,  Figs.  48  and  49  =  38°  40',  and  33°  41'. 
'  48  and  50  =  320. 


The  common  hight  of  the  prismoids  being  h  =  24.  All  the  calcu- 
lations being  carried  out  in  detail ;  all  having  the  same  end  areas, 
320  and  216;  and  all  dimensioned  as  marked  upon  the  figures. 

We  find,  then,  by  all  these  calculations,  the  Solidity  to  be  the  same 
=  3200,  varying  but  a  few  small  decimals,  and  agreeing  with  the 
results  already  ascertained  in  Art.  2. 

This  exhibits  the  equivalency  we  have  been  discussing  (the  figures 
being  quite  unlike),  and  might  readily  be  extended  to  more  compli- 
cated examples,  with  a  like  result. 

15.  Equivalence  of  some  important  Formulas,  for  .computing  the 
Solidity  of  Triangular  Prismoids  of  Earthwork,  contained  within 
Diedral  Angles,  formed  by  Prolonging  the  Side-slope  Planes  to  an  Edge. 

Equivalent  Formulas  are  those  which  reach  the  same  results  by 
unlike  steps — and  in  mathematical  processes  it  is  often  found  that  a 
general  formula  will  hold  in  many  cases,  usually  governed  by  concise 
special  rules,  and  yet  produce  identical  results. 

This  is  equivalency,  and  relates^  in  mensuration  especially  to  the 
Prismoidal  Formula,  which  appears  to  have  a  sort  of  concurrent  juris- 
diction over  the  domain  of  solid  geometry,  along  with  the  special 
rules  for  the  volume  of  each  separate  solid,  producing  exactly  the 
same  results,  though  by  different  steps. 


CHAP.  I.— PRELIM.   PROBS.— ART.  15.  69 

Such  is  particularly  the  case  in  earthwork  solids,  contained  (as 
they  mostly  are)  in  diedral  angles  formed  by  uniform  planes,  called 
side-slopes,  and  having  a  general  triangular  section — two  sides  being 
the  inclined  lateral  planes,  known  as  side-slopes  (continued  to  inter- 
sect for  computation),  and  these  slopes  being  usually  alike  in  inclina- 
tion, while  the  contained  angle  is  equal ; — the  third  side,  or  ground 
line,  alone  being  variable,  and  often  irregular. 

By  geometry,  triangles  having  an  angle  common  or  equal,  and  the 
containing  sides  proportional,  are  similar ;  and  the  areas  of  similar 
triangles  are  always  proportional  to  the  squares  of  any  similar  or 
homologous  lines,  or  to  the  rectangles  of  such  as  have  like  positions 
and  relations  to  each  other : — as  the  squares  of  perpendiculars  from 
the  equal  angles,  or  their  bisectors,  the  rectangles  of  containing  sides, 
the  product  of  hights  and  widths,  etc. 

Now,  these  triangular  sections  of  an  earthwork  solid,  extending 
(for  computation)  from  the  ground  surface  to  the  intersection  of  the 
side-slopes  prolonged  to  an  edge,  are  sections  of  triangular  pyramids,  as 
well  as  of  prismoids ;  and  to  such  solids  the  rules  for  Pyramids,  and 
their  frusta,  as  well  as  the  Prismoidal  Formula,  and  its  modifica- 
tions, apply  concurrently,  and  either  may  be  used  at  will,  with  correct 
results. 

These  considerations  regarding  the  equivalency  of  Pyramidal  and 
Prismoidal  Formulas  in  such  cases  are  important,  and  require  to  be 
well  considered  by  computers  of  earthwork. 

Hutton's  definition  of  the  Prismoid  is  based  on  three  conditions: 

1.  The  two  ends  must  be  dissimilar  parallel  plane  figures. 

2.  They  must  have  an  equal  number  of  sides. 

3.  The  faces,  or  sides  of  the  solid,  must  be  plane  figures  also. 
Usually,  says  Hutton,  the  faces  are  plane  trapezoids. 

Considering,  now,  a  regular  prismoid  as  being  composed  of  known 
elementary  solids. 

Macneill  regards  it  as  formed  of  a  prism,  with  a  wedge  superposed. 
Art.  4  (and  this  is  also  the  case  with  a  frustum  of  a  pyramid,  turned 
upon  its  edge). 

Hutton,  of  two  wedges,  formed  by  a  single  cutting  plane  passed  in 
a  diagonal  direction,  Art.  3. 

The  writer,  as  a  triangular  prism  trebly  truncated,  Art.  1. 

Simpson  (the  father  of  the  prismoid)  gives  no  special  definition, 
but  figures  in  his  work  of  1750  a  rectangular  prismoid  (the  same  or 


70  MEASUREMENT   OF   EARTHWORKS. 

similar  to  that  adopted  and  figured  by  Hutton,  1770);  and  by  a 
single  diagonal  plane,  convertible  into  two  triangular  prismoids. 
(See  Fig.  2.) 

Now,  as  a  triangle  is  the  simplest  of  all  polygons,  so  a  prismoid 
within  a  diedral  angle  (triangular  in  section)  may  be  considered  as 
the  simplest  of  all  prismoids,  though  the  rectangular  prismoid  is 
nearly  so. 

The  simplest  case  of  the  ordinary  trapezoidal  prismoid  of  earthwork 
is  in,  or  upon,  ground  level  transversely. 

In  that  case,  the  cross-sections  are  level  trapezoids,  and  the  solid  is 
obviously  composed  of  a  prism  and  superposed  wedge,  as  in  Macneill's 
solid,  Art.  4. 

Its  volume  may  be  computed  by  Simpson's,  or  by  Button's  general 
rules,  because  this  solid  then  is  strictly  a  prismoid  within  the  scope 
of  Hutton's  definition,  and  as  a  whole  computable  only  by  prismoidal 
rules. 

But  suppose  the  assumed  road-bed  was  taken  less  and  less,  until  we 
reached  the  edge  of  the  diedral  angle,  and  it  became  zero. 

Then,  the  cross-section  from  a  trapezoid  becomes  a  triangle,  and  the 
prismoid  changes  at  once  into  a  fmstum  of  a  pyramid — a  solid  known 
since  the  days  of  Euclid. 

This  solid  becomes  then  computable  by  Euclid's  geometry,  as  the 
frustum  of  a  pyramid — or  by  Equivalent  Level  Hights — by  roots  and 
squares — by  geometrical  average — all  of  which  are  equivalent,  as  are 
the  similar  rules  of  Bidder,  Baker,  Bash  forth,  and  others ;  or,  by 
wedge  and  prism,  by  hights  and  widths  (Simpson),  by  Hutton's  par- 
ticular rule,  by  the  method  of  initial  prismoids,  or,  finally,  by  the 
Prismoidal  Formula  itself,  which  always  holds  alike  for  prismoids, 
pyramids,  or  pyramidal  frusta. 

Hutton  (4to  Mens.,  1770,  p.  155)  shows  that  in  similar  sections  of 
a  pyramidal  frustum  (say  triangular)  the  squares  of  similar  lines,  as 
the  bisector  of  an  equal  angle  (which  the  centre  line  of  a  railroad 
generally  is),  are  as  the  areas  of  the  cross-sections,  or,  conversely,  the 
areas  are  as  the  squares  of  similar  lines  (Chauvenet's  Geom.  iv.  7). 

Then,  from  Hutton's  prob.  7,  cor.  2,  we  have  a  formula  (for  pyra- 
midal frusta)  in  which,  substituting  Bidder's  and  Baker's  notation, 
we  have,  by  a  slight  reduction,  the  identical  rules  given  by  those 
authors  for  the  computation  of  earthwork.* 

*  Bidder,  quoted  in  Dempsey's  Prnc.  Rail.  Eng.,  London,  1855. — Baker,  in  his  Rail- 
way Eng.  and  Earthwork,  London,  1848. 


CHAP.   I.— PRELIM.    PROBS.— ART.    15. 


71 


We  will  now  give  a  diagram  to  illustrate  the  equivalency  of  prismoi- 
dal  and  pyramidal  formulas. 


Fig.  52. 


Road-Tied-lO. 


Fig.  52  represents  the  full  station  of  earthwork,  already  shown  in 
Figs.  22  and  24,  having  a  road-bed  of  18  feet,  and  side-slopes  of  li  to 
1,  with  other  dimensions  as  marked  upon  the  figures. 

Suppose,  in  all  cases  (as  in  Fig.  52),  the  trapezoidal  sections  of  the 
ends  above  the  road-bed  to  be  carried  down  by  prolonging  the  side- 
slopes  to  their  intersection  at  I  I,  the  edge  of  the  diedral  angle. 

(  c  c  =  Top  of  larger  end,  and  h  =  its  hight  =  30  feet. 
Let\bb  =  Top  of  smaller  end,  and  h'  =  its  hight  =  20  feet. 
(  I     =  The  intersection  of  side-slopes,  of  1J  to  1. 

Then,  suppose  a  horizontal  plane  to  be  passed  parallel  to  1 1,  through 
bbb  b,  then  ccbbb  b,  the  part  cut  off,  is  a  wedge,  its  edge  being  b  b, 
the  top  of  the  forward  cross-section  ;  while  h  —  h'  =  the  hight  of  the 
back  c  c  b  b, — and  as  a  wedge  it  may  easily  be  calculated. 

Now,  suppose  the  plane  b  bb  b  moves  downward,  parallel  always  to 
its  first  position  at  the  distance  hf  from  I,  then  the  solid  immediately 
becomes  a  prismoid — being  then  a  prism  with  a  wedge  superposed,  as 
in  Art.  4  (or  analogous  to  it). 

Continue  this  parallel  movement  of  the  plane  downward  until  we 
reach  the  position  a  a  a,  assumed  for  the  road-bed,  and  then  we  have 
the  precise  case  of  A rt.  4 — Sir  John  Macneill's  figure  of  1833.  To 
this  of  course  the  Prismoidal  Formula  applies,  but  the  Pyramidal  For- 
mulas do  not. 

Continue  on  again,  with  the  movement  of  our  supposed  horizontal 
plane  downwards,  until  it  comes  to  I,  I,  (the  junction  of  the  side-slopes), 
then  the  solid  becomes  the  frustum  of  a  pyramid,  triangular  in  sec- 
tion, and  the  wedge  is  absorbed ;  nevertheless,  a  frustum  of  a  pyramid 


72  MEASUREMENT  OF  EARTHWORKS. 

is  also  in  tins  respect  like  unto  a  prismoid,  and  may,  if  we  choose,  be 
regarded  as  a  prism  with  a  wedge  superposed,  and  forming  the  top 
of  the  solid. 

Taking  the  horizontal  plane,  supposed  to  move  parallel  downwards, 
at  three  particular  points  of  its  progress, — at  b,  a,  and  I, — the  calcula- 
tions for  volume  would  be, 

1.  For  the  wedge  alone  =  ccb  bbb 

2.  "     wedge  and  prism,  or  prismoid  =  ccaaabb. 

3.  "     frustum  of  a  pyramid  alone,  both  wedge  and  prism  being 

merged  in  it — and  in  such  case  this  is  the  simplest  and 
best  form  of  calculation,  for  volume. 

We  may  here  remark  that  so  long  as  the  end  cross-sections  contain 
a  road-bed  of  definite  width,  the  solid  is  a  real  prismoid,  and  must  be 
computed  as  such  by  prismoidal  rules  alone;  but  the  moment  the 
angle  at  I  becomes  common  to  both,  then  the  solid  becomes  a  regular 
frustum  of  a  pyramid,  and  all  the  pyramidal  rules  apply,  as  well  as 
the  prismoidal  ones,  to  which  they  are  strictly  equivalent,  whenever  I, 
the  diedral  edge,  is  common  to  both. 

Now,  suppose  the  case  reversed,  and  that  the  horizontal  plane  was 
originally  passed  through  I,  I,  (edge  of  diedral  angle),  and  moves 
gradually  upwards,  parallel. 

At  every  step  of  its  progress,  the  solid,  cut  off  above  I,  is  always  a 
prism,  until  its  limit  has  been  reached,  at  b  b  b  b,  the  top  of  the 
smaller  end — here  the  moving  horizontal  plane  ceases  to  be  longer 
useful  in  illustration;  and  becoming  fixed  at  one  end,  on  the  top  of 
the  far  end  section  as  an  axis,  opens  wider  and  wider  at  the  near  end, 
until  it  attains  the  line  cc  (the  top  of  the  main  solid),  and  completes 
the  wedge  we  have  referred  to,  and  the  pyramidal  frustum  with  it. 

In  this  position  the  whole  solid  is  undeniably  a  prismoid  (if  we 
allow  to  it  an  infinitesimal  road-bed).  So,  also,  it  is  a  frustum  of  a  trian- 
gular pyramid,  both  being  strictly  equivalent,  and  both  computable  by 
the  regular  rules  for  either* 

We  will  now  illustrate  this  equivalence  of  the  Prismoidal  and  Pyra- 
midal Formulas,  in  their  application  to  earthwork  solids,  within 
diedral  angles,  by  a  few  examples. 

Taking  the  dimensions  of  Figs.  22  and  24,  with  1£  to  1  side-slopes, 
and  road-bed  of  18,  for  thfe  numbers  to  be  employed — the  diedral 
angle  being  common  to  both. 

*  As  might  be  inferred  from  Button's  remarkable  chapter  on  the  Cubature  of  Curves 
(4to  Hens.,  1770). 


CHAP.    I.—  PRELIM.   PROBS.—  ART.   15.  73 

1.  Priwnoidally.  —  By  the  direct  and  cross  multiplication  of  Hights 
and  Widths.     Formula  at  the  end  of  Art.  9  ......     VIII. 

TT.         (  h  =  30  \s  w  =  90  )  w.  ,,,  „ 

Hights  |  /t,  =  20  X  u''  =  60  j  Wldths- 


30        20  30  90  2700 

90        60  60  20          1200 

2700     1200    2)1800  +  1800 

6)5700 


950  X  100  =-  95000  = 
Solidity,  as  before  computed. 


2.  Pyramidally.  —  By  the  rules  of  Baker's  Earthwork. 


30 
30 

900 

r 
I 

20    30 
20    20 

400   600 

=  H 
=  100 

900 
400 
600 

1900 
50 

95000 

3)150 
~~50 

Solidity,  as  before  computed 


3.  Prismoidally. — By  Simpson's  rule,  modified  for  triangular  solids. 

Ilights.        Widths. 

30  X     90     =     2700 
20  X     60     =     1200 


Sums,  50  X   150     «=     7500 
12)11400 


950  X  100  =  95000  =  Solidity,  as 
before  computed. 


4.  Pyramidally. — By  Roots  and  Squares,  Art.  10  (c). 

End  Areas  .     .  =  1350         600 
Roots .  .  =      36-74      24-50 


Sum    ....==         61-24 
Square  of  Sum  =     3750 
End  Areas.     .  =  {  ^ 

6)5700 


950  X   100  =  95000  =  Solidity,  as 
before  computed. 


74  MEASUREMENT  OF  EARTHWORKS. 

5.     Finally,  by  Warner's  Earthwork,  Art.  112. 

Hts.  Wds. 

Difference  =  10  j  ^  x     60  }  Difference  =  30- 

Sums         .  50  X  150    .  .  =    7500 


937-5  =  1st  term. 


X  100  =  95000     =  Solidity. 


So,  we  may  safely  assume  that  the  Pyramidal  Formulas  of  Bidder, 
Baker,  and  others,  the  Geometrical  Average,  Equivalent  Level  Rights, 
Euclid's  rule  for  the  frustum  of  a  pyramid,  etc.,  are  all  strictly  equiva- 
lent to  the  Prismoidal  Formula,  and  its  modifications,  when  applied  to 
earthwork  solids,  within  diedral  angles,  —  on  ground  transversely  level. 

16.  Summary  of  Rules  and  Formulas  from  the  Preliminary  Problems. 

It  will  be  found  convenient  to  use,  substantially,  the  same  notation 
for  the  Prismoidal  Formula,  and  its  numerous  modifications,  wher- 
ever practicable. 

b   =  Base,  or  area  of  end  assumed  for  such. 
t    =  Top,  or  area  at  the  other  end. 

Thus  let^m  =  Hypothetical  Mid-section,  used  in  computation. 
h   =•  Length  or  hight  of  the  Prismoid. 
S  =  Solidity  or  volume. 

Then,  the  Prismoidal  Formula  can  always  be  in  substance  expressed 
by—  —  ^  —  -  X  h  —  S,  when  a  mean  area  is  desired,  or  by 

(b  -f  4  m  -\-  f)  X  i  h  —  S,  for  rectangular  prismoids,  or  equivalent 
solids;     or,    when    triangular    prismoids    are    under    computation, 

2  b  +  2  t  f  8  ?TI 

-----  X  h  =  fe,  equivalent  in  using  triangular  sections 

and  double  areas,  to  this  rule  in  words  :  The  separate  products  of  hights 
by  widths  at  each  end,  plus  product  of  sums  of  hights  and  widths  at  both 
ends,  and  the  sum  of  these  three  products,  multiplied  by  ^  h  =  Solidity. 
The  following  modification  of  this  rule  may  be  sometimes  useful  in 
computing  the  volume  of  triangular  earthwork  solids  :  The  products 
of  the  direct  multiplication  of  hight  by  width  at  each  end,  plus  sum  of 
half  products  of  the  cross  multiplications  of  alternate  hights  and  widths  a*, 


CHAP.   I.— PRELIM.    PROBS.— ART.   16. 


75 


both  ends,  multiplied  by  £  h  =  solidity  from  ground  to  intersection  of 
slopes,  and  mimis  the  grade  prism  =  solidity  from  road-bed  to  ground. 

Many  other  expressions  are  assumed  for  special  purposes  by  the 
Prismoidal  Formula ;  but  no  matter  into  what  shape  it  be  transformed, 
the  essential  idea  must  always  be  borne  in  mind  that  this  formula,  in 
words,  concisely  is, 

"  The  sum  of  the  areas  of  the  two  ends,  and  four  times  the  sec- 
tion in  the  middle,  multiplied  into  £  h  =  S."     (Hutton,  1770.) 

Such  is  the  simple  expression  of  this  celebrated  formula — given  a 
century  ago — which  applies  not  only  to  all  prismoids,  but  to  all  right- 
lined  solids,  and  many  curved  ones  too.* 


SUMMARY. 

For  rectangular  prismoids,  or  any  prismoid,  reduced 
to  an  equivalent  rectangular  section,  we  have  Simp- 
son's original  rule  expressed  by  sides  of  the  end  rect- 
angles, referring  to  Fig.  2,  Art.  2.     But  it  is  more 
convenient,    perhaps,  for   our   purpose,  to    designate 
these  sides  relatively,  as  hights  and  widths,  and  in  this 
form  we  ma^  write  Simpson's  rule  as  follows  : 

(Hight  X  Width  of  one  end)  +  (Right  X  Width 
of  other  end)  -f  (Sum  of  Hights  X  Sum  of  Widths 
of  both  ends)  X  £  h  ==  S. 

And  the  transformation  of  this  formula,  for  use  in 
the  computation  of  triangular  prismoids  (like  earth- 
work), placing  it  in  Button's  form. 

—  TViq   TVTpflll   A»*OQ    art  A  N/  ^  —  .  Ks\7frJ*fai 

Article.  I  Formula. 

2. 

2. 
3. 
3. 

I. 

II. 
III. 
IV. 

±2 

For  rectangular  prismoids,  considered  as  two  wedges. 
We  have  Hutton's  General  Rule  for  any  prismoid, 
(6  +  <  +  4m)X/> 

6 
We  have  also  Hutton's  Particular  Rule. 

(2L  +  1  X  B  +  21  +  L  X  6)  X  $h  =  S. 

*  The  English  engineers  have  for  many  years  unhesitatingly  applied  this  formula  to 
the  warped  solids  of  earthwork.  See  Dempsey's  Practical  Railway  Engineer,  4th  edition, 
4to,  London  (1855),  pp.  71  to  74.  And  in  this  country,  Prof.  Gillespie  (1857),  and  John 
Warner,  A.  M.  (1861),  have  also  discussed  the  subject  of  Warped  Solids  of  Earthwork. 


76 


MEASUREMENT  OF  EARTHWORKS. 


Article. 

Formula. 

SUMMARY—  Continued. 

3. 

V. 

For  unusual  and  irregular  prismoids  we  have  the 

method  of  "  Initial 

Prismoids"  deduced  from  Hutton. 

6. 

VI. 

For  a  prismoid, 

composed  of  a  prism  and  wedge, 

superposed. 

(B  +  b  -f  b)  X  (H 

1      (  n    w           fipvnfir*  f'vinnn'lo^   \S 

6 

7. 

VII. 

For  a  trapezoidal  prismoid  of  earthwork,  taken  as 

two  wedges. 

We  have  the  following  Rule  : 

Add  road-bed   +   top-  width  -f- 

In  1st  cross-section 

road-bed  of  2d  section  ;  multiply  the 
sum  of  these  three  by  level  hight 

of  section,  and  reserve  the  product. 

Add  road-bed  -f  top-width  -f  top- 

In  2d  cross-section 

width  *of  1st  section;  multiply  the 
sum  of  these  three  by  level  hight 

of  section,  and  reserve  the  product. 

Finally,  add  the 

two  products  reserved,  and  £  of 

their  sum  is  the  mean  area  of  the  Prismoid,  which, 

multiplied  by  length  =  Solidity. 

For  a  triangular  prismoid  of  earthwork,  we  have 

the  following  modification  of  the  Prismoidal  Formula, 

operating  by  direct  and  cross-multiplication  of  hights 

' 

and  widths.     All 

hights  being  taken  at  centre  from 

ground  to  intersection  of  slopes,  and  all  widths  from 

top  to  top  of  slopes  on  both  sides  of  centre. 

Let  h  and  hr  = 

the  hights.  w  and  wf  =  the  widths. 

Then, 

Hights.    Widths.       \ 

h    X  iv 

h  wr  -4-  Tir  w 

9. 

VIII. 

X 

h'  X  w' 

/>  -)/,  _i_  //  /!//  _L_ 

,and                       6 

Length  =  100, 

length  =  S. 

usually.          / 

Article 


Formula. 


CHAP.  I.— PRELIM.   PROBS.— ART.  16.    - 
SUMMARY—  Continued 


10. 


10. 


IX. 


10. 


XI. 


Simpson's  Rule,  for  the  Quadrature  and  Cubature 
of  Curves  (adopted  by  Hutton),  and  copied  from  the 
4to  Mens.  (1770). 

Sum  extreme  ordinates  =  A.  "| 
"    all  even        "         =  B.   [A-f  4B  +  2C 
"    all  odd          "         =  C.   [          ~3~~ 
Common  distance  =  D.  J  D  =  area  or  solidity. 

For  convenience  we  may  transform  this  into, 

— X  2  D  =  area  or  solidity. 

To  find  the  solidity  of  a  triangular  prisraoid  by 
roots  and  squares. 

'  h  and  h'  =  The  end  hights  or  representative 
square  roots  of  the  areas  of  the  ends  (between 
ground  and  intersection  of  slopes),  at  regular 
stations,  numbered  even. 

m  —  Place  of  mid-section,  represented  by  its  ordi- 
nate,  and  numbered  odd. 

Length  =  Usually,  100,  between  principal  sta- 
tions. 

&'  -f  fr'2  +  (h  +  hj  vx  , 

— |^  -  X  length  =  S. 

"Which,  for  one  station,  is  equivalent  to  Hutton's 
rule  above.  This  is  a  very  important  transformation 
of  the  Prismoidal  Formula,  and  should  be  well  con- 
sidered, with  the  examples  in  Art.  1O. 

One  of  the  earliest  followers,  in  the  path  projected 
by  Sir  John  Macneill,  of  using  the  Prismoidal  For- 
mula, with  auxiliary  tables,  for  correctly  computing 
the  volume  of  earthwork  solids,  was  G.  P.  Bidder, 
C.  E.,  who  adopted  the  obvious  plan  of  imagining  the 
side-slopes  to  be  moved  parallel  inward,  to  intersect  at 
grade,  and  then  computing  the  triangular  solid  thus 
formed  as  a  prismoid,  or  the  frustum  of  a  pyramid 
(both  being  equivalent  in  these  circumstances) ;  finally, 
calculating  the  centre  part  (or  core)  as  a  prism  sepa- 
rately, and  adding  the  two  for  the  volume  of  the  whole. 
The  core  being  computed  for  one  foot  wide  only, 


MEASUREMENT   OF  EARTHWORKS. 


Article. 

Formula. 

SUMMARY—  Continued. 

and  then  multiplied  by  the  width  of  road-bed  intended 
to  be  given.*  (This  is  the  plan  of  Macneill's  second 
series  of  Tables,  for  various  side-slopes,  and  base  of 
one  foot.) 

Bidder's  formula  for  the  slopes  united  is,  [  (a  -f  &)2 
—  a  5]  ||  =  S,  in  cubic  yards  for  a  66  foot  chain,  a 
and  b  being  the  hights  or  depths  at  the  ends. 

This  is  identical  with  the  formulas  of  Baker,  Bash- 
forth,  and  others,  of  subsequent  writers  :  =  (a2  -f  a  b 
+  6'2)  II  =  S,  in  cubic  yards,  and  is  in  fact  the  alge- 
braic expression  for  the  volume  of  the  frustum  of  a  tri- 
angular pyramid,  demonstrated  in  all  the  elements  of 
geometry  —  supposed  to  have  been  originated  by  Euclid 
(about  300  B.  c.),  and  known  in  this  country  as  the 
method  of  Geometrical  Average. 

These  formulas  are  equivalent  to  the  following,  men- 
tioned in  Art.  12. 

(Sum  of  sqs.  of  hts.)  -f  (Sq.  of  sum  of  hts.)        ^       a 

12. 

XII. 

6 

_  2  (Sum  sqs.)  -f  2  (Rect.  of  hights) 

(Sum  sqs.  of  hights)  +  (Rect.  of  hights)  w  ^       Q 

0                                                                X   "-            k, 

which,  for  a  four  pole  chain,  and  cubic  yards,  becomes 
equivalent  to  the  formulas  above,  by  introducing  the 
proper  fractional  multipliers  —  the  hights  are  the  square 
roots  of  the  areas. 

*  A  similar  plan  of  computing  and  tabulating  the  slopes  and  core 
separately  :  the  latter  on  a  base  of  unity,  to  be  subsequently  multi- 
plied, by  any  road-bed,  is  also  that  of  E.  F.  Johnson,  C.  E.  —  the 
pioneer  of  Earthwork  Tables  in  this  country  (New  York,  1840)  —  and 
has  been  followed  by  several  other  writers;  indeed,  it  is  a  method  so 
obvious  as  to  be  likely  to  occur  to  any  student.  This  core  and  slope 
method  originated  by  Bidder  and  Johnson  (some  30  years  ago),  and 
since  repeated  by  numerous1  writers,  is  now  again  reiterated  by  the 
latest  compiler  of  Earthwork  Tables,  E.  C.  Rice,  C.  E.  (St.  Louis, 
Mo.,  1870). 

CHAPTER  II. 

FIRST  METHOD  OF  COMPUTATION  BY  MID-SECTIONS,  DRAWN  AND 
CALCULATED  FOR  AREA,  ON  THE  BASIS  OF  BUTTON'S  GENERAL 
RULE. 

17 Since  1833 — the  date  of  publication  of  Sir  John  Macneill's 

meritorious  volume  on  the  mensuration  of  earthworks,  for  canals, 
roads,  and  railroads — the  investigations  of  numerous  able  writers  in 
various  countries  have  shown,  conclusively,  that  the  Prismoidal  For- 
mula (adopted  by  Macneill)  furnishes  the  most  convenient,  if  not  the 
only  correct  rule  for  the  measurement  of  the  immense  bodies  of  mate- 
rial employed  in  earthworks,  and  removed  from,  or  supplied  to,  the 
irregularities  of  the  ground  encountered  by  the  location  of  lines, 
under  the  general  name  of  excavation  or  embankment. 

The  writer,  as  long  ago  as  1840,  in  the  Journal  of  the  Franklin 
Institute  of  Pennsylvania,  repeated  the  demonstration  of  the  formula 
referred  to,  by  means  of  a  simple  figure,  and  established  its  connection 
with  the  ordinary  rules  for  the  volume  of  the  three  principal  right- 
lined  bodies,  known  to  solid  mensuration — the  Prism,  Wedge,  and 
Pyramid — (to  all  of  which,  whether  complete  or  truncated,  the  Pris- 
moidal Formula  correctly  applies) ;  these  are  the  elementary  solids 
which  enter  into  the  composition  of  a  station  of  earthwork,  and  sepa- 
rately, or  together,  are  all  computable  by  the  same  rule. 

He  also  showed,  by  numerous  examples  (worked  out  in  detail)  of 
the  leading  forms  assumed  by  railroad  earthworks,  that  by  means  of 
hypothetical  mid-sections,  deduced  from  the  usual  cross-sections  taken 
in  the  field  (and  diagrammed  between  them  if  necessary),  the  volumes 
of  excavation  and  embankment  solids  could  be  computed  correctly 
without  unusual  labor,  and  with  more  than  usual  accuracy.  This 
method  was  made  to  depend  essentially  upon  two  points :  * 

*  Journal  of  the  Franklin  Institute  (Philadelphia,  1840). 

79 


gO  MEASUREMENT  OF  EARTHWORKS. 

1.  "That  the  formula  expressing  the  capacity  of  a  prismoid  is 
the  fundamental  rule  for  the  mensuration  of  all  right-lined  solids, 
\vhose  terminations  lie  in  parallel  planes,  and  is  equally  applica- 
ble to  each." 

2.  "  That  any  solid  whatever,  bounded  by  planes,  and  parallel 
ends,  may  be  regarded  as  composed  of  some  combination  of 
prisms,  prism oids,  pyramids,  and  wedges,  or  their  frusta,  having 
a  common  altitude,  and  hence  capable  of  computation  by  the  gen- 
eral rule  for  prismoids." 

All  excavation  and  embankment  solids  come  within  the  scope  of 
these  definitions,  and  all  are  computable  with  ease  and  accuracy  by 
means  of  the  Prismoidal  Formula. 

These  views  have  met  with  general  acceptance  from  most  practical 
writers,  but  many  useful  transformations  and  modifications  have 
naturally  been  indicated  ;  all  grounded  upon  the  same  formula  which 
appears  to  have  originated  with  THOMAS  SIMPSON,  an  eminent  mathe- 
matician, and  was  demonstrated  and  published  by  him  (for  rectangular 
prismoids)  in  London,  1750  (Arts.  1  and  2),  but  generalized  and 
made  more  useful  by  HUTTON,  in  1770  (Art.  3). 

This  extraordinary  formula  is  not  only  the  fundamental  rule  for 
all  right-lined  solids,  but  reaches  also  to  many  curved  bodies  and 
warped  surfaces  (as  before  mentioned),  so  that  it  may  safely  be 
assumed  as  correct  for  all  the  earthwork  solids  in  common  use,  which, 
indeed,  are  invariably  laid  out  with  the  view  of  reducing  the  ground, 
however  irregular,  to  equivalent  planes  (as  near  as  may  be),  by  means 
of  leyels  and  sections,  taken  at  short  distances ;  and  though  this  effort 
may  not  be  entirely  successful  in  practice,  it  must  be  so  nearly  so  that 
the  warped  surfaces,  remaining  involved  in  the  solid,  can  only  differ 
slightly  (if  at  all)  from  those  for  which  the  Prismoidal  Formula  is 
known  to  hold. 

As  a  general  rule,  it  may  therefore  be  considered  as  close  an 
approximation  to  existing  facts  as  is  admitted  by  any  convenient 
method  within  the  present  range  of  human  knowledge,  and  far  more 
accurate  than  any  of  the  proximate  rules,  which  have  been  extensively 
employed  for  the  solution  of  the  complicated  problems  of  earthwork. 

As  a  preliminary  matter,  it  is  necessary  now  to  make  some  remarks 
on  the  manner  of  collecting  data  in  the  field,  for  subsequent  use  in 
calculating  the  quantities  of  earthwork  solids. 

The  centre  or  guiding  line  of  the  road  or  work  having  been  care- 
fully located  upon  the  ground,  and  marked  off  in  regular  stations — 


CHAP.  II.—  FIRST  METH.  COMP.—  ART.  17.  81 

usually  of  one  hundred  feet  each  —  the  next  operation  is  to  cross-section 
the  work,  with  level,  rod,  and  tape;  most  engineers  also  using  the 
clinometer,  or  slope  level,  as  an  auxiliary,  in  some  stages  of  the  pro- 
cess. The  centre  line  is  assumed  in  all  cases  to  be  straight,  from  point 
to  point,  and  generally  to  be  a  tangent  line,  to  which  the  cross-sec- 
tions are  perpendicular,  but  owing  to  the  convergence  of  the  radii 
upon  curves,  this  is  not  strictly  correct  —  though  within  the  limits  of 
the  work  staked  out,  that  convergence  is  but  slight  ;  nevertheless,  the 
cross-sections  (before  proceeding  to  level  them)  should  be  set  out 
approximately,  normal  to  the  tangents,  and  radial  to  the  curves  ;  and 
upon  all  curves,  or  at  least  on  all  of  small  radius,  intermediates  at  half 
distance  should  be  placed,  or,  if  the  curves  are  unusually  sharp,  even 
at  the  quarter  of  a  regular  station. 

Some  engineer  manuals  furnish  formula  for  the  correction  of  quan- 
tities upon  curved  lines,*  but  they  are  rarely  used  ;  a  simple  reduction 
of  distance  between  the  cross-sections,  or  a  closer  assemblage  of  them, 
being  usually  deemed  sufficient. 

The  surface  of  the  ground  f  is  regarded  by  the  engineer  as  being 
composed  of  planes  variously  disposed,  with  relation  to  each  other,  so 

*  The  simplest  and  most  convenient  rule  for  this  purpose,  is  that  of  Warner's  Earth- 
work (1861).  This  rule  has  been  adopted,  and  somewhat  simplified,  by  Prof.  Rankine, 
in  Useful  Rules,  etc.  (London,  1866). 

The  process  is  :  First,  to  calculate  the  solidity  of  the  earthwork  to  the  intersection  of 
the  slopes  (as  though  the  line  were  straight),  and  then  to  multiply  it  by  a  factor,  which 
corrects  for  curvature. 

Difference   slope  distances   , 

This  factor  is  found  thus  :   -  -  -  -  'jbl«     The  corrective  quotient 

6  Radius  of  curve. 

being  added  to  unity,  when  the  greater  slope  distance  lies  outward  from  the  curve,  or 
subtracted,  if  otherwise. 

For  example,  take  a  curve  of  700  feet  radius,  lying  upon  a  heavy  embankment,  along 
a  ground  surface  sloping  uniformly  inwards,  towards  the  centre  of  the  curve,  at  the  rate 
of  15°.  The  road-bed  being  24  feet  wide,  and  side-slopes  1  J  to  1. 

Let  the  difference  of  slope  distances  be  42  feet,  the  greater  being  inwards,  and  suppose 
the  whole  volume,  for  straight  work  =  5917  cubic  yards  to  intersection  of  slope.  Then, 

=  —  -02,  and  1  —  -02  =  -98,  the  factor  required.     Then,  5917  X  *98 


6  X  <  "" 

cubic  yards,  and  5799  —  grade  prism  (356)  =  5443  cubic  yards,  the  volume,  corrected  for 
curvature.  The  difference  in  this  case,  produced  by  the  curvature  of  the  line,  being  118 
cubic  yards,  for  the  station  computed. 

The  correction  for  other  curves  would  be  inversely  as  their  radii,  and  for  a  1°  curve, 
similarly  situated,  about  15  cubic  yards,  per  station. 

The  difference  of  the  distances  out  from  the  centre  are  the  same  thing  as  Prof.  Ran- 
kine's  difference  of  slope  distances  —  since  the  former  involve  an  equivalent  quantity  on 
both  sides  of  centre,  equal  to  half  the  road-bed. 

f  Journal  Franklin  Institute  (1840). 
6 


82  MEASUREMENT   OF  EARTHWORKS. 

that  any  vertical  section  will  exhibit  a  rectilineal  figure,  more  or  less 
regular.  This  supposition,  though  not  strictly  correct,  is  sufficiently 
accurate  for  practical  purposes. 

Upon  the  cross-sections  (taken  near  enough  together  to  define  posi- 
tively the  general  figure  of  the  surface),  sufficient  level  points  are 
obtained  transversely,  by  level  and  rod,  their  distances  out  from 
centre  being  simultaneously  measured,  with  a  tape  line ;  in  this  man- 
ner, both  vertically  and  horizontally,  in  relation  to  established  planes, 
the  position  of  all  the  points  necessary  to  determine  the  configuration 
of  the  ground  is  well  ascertained. 

These  points  of  elevation,  or  depression,  are  commonly  called  plus 
or  minus  cuttings  (or  simply  cuttings),  and  the  horizontal  distances 
which  fix  their  relation  to  the  centre  are  shortly  called  distances  out. 

The  details  of  the  operation  Staking  the  cuttings,  or  cross-sectioning 
the  work  (a  matter  of  vital  importance  in  correct  measurement), 
require  good  judgment  and  accuracy ;  but  are  so  well  known  to  prac- 
tical engineers  as  to  render  unnecessary  a  description  at  length.  This 
operation,  however,  is  the  absolute  foundation  upon  which  the  whole 
fabric  of  computation  rests,  and  if  it  be  not  judiciously  executed,  all 
rules  are  vain. 

We  may  here  mention  a  general  maxim,  which  should  never  be 
neglected,  if  accurate  results  are  desired,  viz. :  At  every  change  of  sur- 
face  slope,  transversely,  single  cuttings  and  distances  out  must  be  taken ; 
and  at  every  longitudinal  change,  sections  of  cuttings,  or  cross-sections. 

Upon  very  rough  ground  it  is  customary  to  make  the  lateral  dis- 
tances apart  of  the  cuttings,  uniformly  10  feet,  which  materially 
facilitates  the  subsequent  calculations  ;  so  much  so,  indeed,  that  on  a 
rock  side  hill  it  is  often  advisable  to  use  this  distance,  even  though 
the  ground  seems  not  actually  to  need  it;  the  cuttings  and  distances 
out  are  commonly  taken  in  feet  and  tenths,  and  the  regular  stations 
of  one  hundred  feet  are  subdivided  by  cross-sections  into  shorter 
lengths,  if  the  ground  requires  it,  as  is  frequently  the  case.  One  foot 
being  usually  the  unit  of  linear  measure,  one  hundred  feet  a  regular 
station,  and  the  cubic  yard  the  unit  of  solidity,  in  earthwork. 

Though  not  indispensably  necessary,  it  will  be  found  convenient 
in  using  the  prismoidal  method  of  calculation,  as  well  as  conducive 
both  to  expedition  and  accuracy,  to  observe  the  following  rules  in 
"taking  the  cuttings,"  as  far  as  the  character  of  the  surface  will 
admit,  viz. : 


CHAP.    II.— FIRST   METH.  COMP.— ART.  17.  83 

1.  On  side-hill,  at  each  cross-section,  where  the  work  runs 
partly  in  filling  and  partly  in  cutting,  ascertain  the  point  where 
grade,  or  bottom,  strikes  ground  surface. 

2.  On  every  cross-section,  take  a  cutting  at  both  edges  of  the 
road,  or  at  the  distance  out  right  and  left  of  one-half  the  base. 

3.  Always  take  a  cross-section,  whenever  either  edge  of  the  road- 
bed strikes  ground  surface,  and  set  a  grade  peg  there  to  guide 
the  workmen. 

4.  On   rough  side-hill,  or  wherever   the  ground   appears  to 
require  it,  take  the  cuttings  (not  otherwise  provided  for)  at  ten 
feet  apart. 

5.  Wherever  the  ground  admits,  place  the  cross-sections  at 
some  decimal  division  of  100  feet  apart,  as  10,  20,  30,  etc. 

6.  Endeavor  to  take  the  same  number  of  cuttings,  in  each 
*        adjacent  cross-section,  to  facilitate  the  computation. 

7.  On  plain  and  regular  ground,  take  three  cuttings  only — at 
centre  and  both  slopes. 

If  these  simple  directions  are  observed  by  the  field  engineer,  and 
the  work  carefully  done,  much  labor  will  be  saved,  both  to  him,  and 
to  the  computer  in  the  office. 

In  all  cases  of  side-long  ground,  we  suppose  it  to  slope  in  the  same 
general  direction,  between  the  end  sections,  and  do  not  admit  of  oppo- 
site surface  slopes,  because,  under  the  general  rule,  the  field  engineer 
would  place  a  cross-section  at  the  point  of  change  slope,  and  render 
the  consideration  of  opposite  slopes,  and  the  warped  surfaces  they 
always  produce,  entirely  unnecessary ;  indeed,  by  more  closely  assem- 
bling the  cross-sections  together,  we  can  practically  reduce  even  the 
most  irregular  surface  to  a  series  of  planes  coincident  with  it. 

Nevertheless,  an  able  writer  *  has  shown  that  warped  solids  of  a 
certain  kind  are  computable  by  his  rules ;  and  the  late  Professor 
Gillespie,  in  several  valuable  essays,  has  demonstrated  that  hyper- 
bolic paraboloids  at  least  could  be  correctly  calculated  by  the  Pris- 
raoidal  Formula ;  while  English  engineers  have  long  used  this  rule 
for  computing  the  volume  of  earthwork  solids,  ivith  warped  surfaces;^ 
it  appears,  however,  to  be  more  certain  and  satisfactory  if  we  confine 
the  operations  of  this  formula  to  solids  bounded  by  plane  surfaces  as 
nearly  as  circumstances  admit;  but  it  is  fortunate  that  our  rule  is 

*  John  Warner,  A.  M.,  Computation  of  Earthwork  (1861). — Prof.  Gillespie,  Manual 
of  Roads  and  Railroad?,  10th  edition  (1871). 

f  Dempsey,  Practical  Railway  Engineer  (London,  1855). 


84  MEASUREMENT  OF  EARTHWORKS. 

known  to  hold  for  some  descriptions  of  warped  ground,  and  hence  can 
hardly  fail  to  proximate  results,  near  unto  the  truth,  however  much 
the  surface  may  be  warped,  between  the  cross-sections,  if  they  have 
been  judiciously  placed  by  the  field  engineer. 

a  .......  The  modification  of  the  Prismoidal  Formula,  which  we 

shall  employ  in  this  first  method  of  computation,  will  be  that  designed 
to  find  a  mean  area,  to  be  subsequently  employed  by  the  aid  of  our 
Table,  at  the  end,  to  ascertain  the  cubic  yards  of  volume. 

This  formula  comes  from  that  generalized  by  Hutton  (1770)  through 
the  special  mid-section,  and  is  expressed  in  the  beginning  of  Art.  16 
as  follows  :  * 


Summarily  expressed  in  words  as  follows;  One-sixth  the  sum  of 
end  areas,  and  quadruple  mid-section,  multiplied  by  length,  gives  the 
Solidity. 

This  general  formula  (identical  with  one  of  Hutton's)  requires  three 
areas  (one,  the  mid-section,  deduced  from  the  others),  and  also  the 
hight  or  length  of  the  Prismoid  to  be  given;  and  by  its  aid  we  pro- 
pose in  illustration  to  furnish  five  examples  of  calculation. 

1.  Of  a  regular  station,  of  three-level  ground. 

2.  Of  the  same  length,  of  five-level  ground. 

3.  Of  seven-level  ground. 

4.  Of  nine-level  ground. 

5.  Of  a  portion  of  excavation  and  of  embankment  adjacent, 
with  an  oblique  passage  between  them,  from  one  to  the  other. 

We  here  follow  a  classification  of  ground  nearly  resembling  that 
adopted  by  the  late  Prof.  Gillespie  (one  of  our  ablest  writers  upon 
earthwork),  who  enumerates  four  classes  only,  under  the  simple 
nomenclature  of,  1,  one-level;  2,  two-level;  3,  three-level;  4,  irregular 
ground;  and  under  these  four  classes,  he  dealt  with  the  problems  of 
earthwork  in  his  excellent  lectures  "  to  the  Civil  Engineering  Classes 
in  Union  College."  f 

*  "  This  rule,"  says  Prof.  Rankine,  in  Useful  Rules  and  Tables,  2d  edition,  London, 
1867,  p.  74,  "  applies  generally  to  any  solid  bounded  endwise  by  a  pair  of  parallel  planes, 
and  sideways  by  a  conical,  spherical,  or  ellipsoidal  surface,  or  by  any  number  of 
planes." 

j-  Manual  of  Roads  and  Railroads,  10th  edition  (1871). 


CHAP.  II.— FIRST  METH.  COMP.— ART.  18.  85 

We  think,  however,  that  few  engineers  would  be  willing  to  class 
ordinary  five-level  ground  as  irregular ;  for  such  ground  would  in  fact 
be  produced  simply  by  the  angle  levels  commonly  taken,  which  at 
once  convert  the  plainest  three-level  into  five-level  ground. 

But  ground  requiring  more  than  five  cuttings  on  one  cross-section, 
all  would  probably  agree  in  classifying  as  irregulary  aiid  such  is  the 
view  taken  by  the  present  writer. 

This  would  bring  all  ground  whatever  within  the  scope  of  five 
classes,  and  make  but  a  slight  variation  in  Gillespie's  nomenclature. 
1.  Level  ground,  where  the  centre  cutting  alone  is  sufficient  for  vol- 
ume. 2.  Ground  slightly  inclined,  where  side-high ts  only  may  have 
been  taken.  3.  Ordinary  ground,  requiring  centre  and  side-hights. 
4.  Same  as  3,  with  the  addition  of  angle  levels,  or  one  cutting  right 
and  left  of  centre,  besides  those  at  the  slope  stakes.  5.  Irregular 
ground, — such,  or  any  similar  classification  would  somewhat  simplify 
the  matter  of  earthwork,  but  it  is  not  indispensable.  Centre  cuttings, 
or  level  bights  at  the  centre,  are,  however,  invariably  taken  in  the 
field,  and  recorded  at  the  time,  whether  they  be  subsequently  used  or 
not,  so  that  class  2  would  seldom  occur  on  original  ground. 

The  method  of  measuring  the  capacity  of  long  irregular  solids,  by 
means  of  normal  sections,  at  short  distances,  has  long  been  used  by 
mathematicians  ;  of  which  numerous  examples  may  be  found  in  Hut- 
ton  (1770),  as  well  as  in  the  demonstration  and  use  of  Simpson's  rule 
for  quadrature  and  cubature,  referred  to  in  many  works,  both  civil 
and  military. 

This  method  then  was  naturally  adopted  by  the  earlier  engineers 
for  the  mensuration  of  earthwork,  and  has  been  continued  down  to 
the  present  day  with  little  chance  of  being  superseded  ;  as  the  areas 
of  the  sections,  commonly  known  to  the  engineer  as  cross-sections,  are 
not  only  useful  in  the  computation  of  solidity,  but  also  in  many  other 
ways,  during  the  progress  of  earthworks  ;  and  consequently  those  rules 
which  disregard  the  areas  of  cross-sections,  and  aim  directly  at  the 
volume  alone  of  excavation  and  embankment,  are  less  useful  (even  if 
more  concise}  than  those  which  require  the  sectional  areas  to  be  first  com- 
puted. 

18.  Examples  in  Computation  by  the  First  Method. 

In  computing  by  this  method,  the  Grade  Prism  is  not  required,  and 
is  not  used,  but  it  may  be  employed  in  verification. 

Example  1. — We  will  now  give  three  figures  (Figs.  53,  54,  and  55), 
representing  three  cross-sections,  upon  one  regular  station  of  100  feet 


86 


MEASUREMENT  OF  EARTHWORKS. 


in  length,  of  a  railroad  cut  with  side-slopes  of  1  to  1,  and  road-bed  of 
20  feet — the  other  dimensions  being  as  marked  upon  the  figures. 

In  these,  the  first  and  last  represent  the  end  cross-sections  of  the 
100  feet  station,  supposed  to  have  been  regularly  taken  in  the  field. 

The  other  (Fig.  54)  being  the  hypothetical  mid-section,  deduced  from 
the  end  ones,  as  required  by  HUTTON'S  General  Rule. 


Bg.53 


~ .»..._  — 3-°- "/ 


t 


These  cross-sections  are  marked  as  follows: 


b   =  890  Area. 
m  =  625     " 
t    =  400     " 
Length,  100  feet 


1 


Example  1. 


CHAP.  IL— FIRST   METH.  COMP.— ART.  18. 


87 


And  the  calculations  for  solidity  are  as  below: 

890  =  b. 

400  =  t. 

2500  =  4  in. 
Calculations, 


631'7  =  Prismoidal  Mean  Area. 


2339-6  =  Cubic  Yards  (by  Table)  for  100  feet. 

The  above  example. is  for  plain  ground  of  "three  levels"  as  classed 
by  Professor  Gillespie. 

Example  2. — We  will  now  give  an  example  of  a  railroad  cut,  with 
the  same  road-bed  (20)  and  ratio  of  side-slopes  (1  to  1),  in  five-level 
ground. 


The  three  cross-sections,  upon  the  regular  station  of  100  feet,  are 
numbered,  Figs.  56,  57,  and  58,  and  marked  b,  m,  and  t,  the  middle 


88 


MEASUREMENT  OF  EARTHWORKS. 


one  being  Hutton's  hypothetical  mid-section,  deduced  by  Arithmetica\ 
Averages  from  b  and  t,  the  cross-sections,  assumed  to  have  been  taken 
in  the  field,  with  rod,  level,  and  tape,  in  the  usual  manner. 

Cross-sections. 
b    =  244  Area. 
Example  2     m  =  286     " 
t    =  331     " 
Length  100  feet  =  h. 

And  the  calculations  for  solidity  are  as  follows : 

244     =  b. 

1144     =  47M. 

331     =  t. 

6)1719 

286*5  =  Prismoidal  Mean  Area. 
And  for  Cubic  Yards,  in  100  feet  long,  per  Table  =  1061-1. 

Example  3. — We  will  now  give  an  example  of  a  railroad  cut,  simi- 
lar to  the  preceding,  base  20,  slope  ratio  r  =  1,  in  seven-level 
ground. 

Cross-sections  and  areas. 

b    =  524 

m  =  537 

*    =  551 

Length,  100  feet  =  h. 


Example  3 


Calculations  for  solidity : 

524     =  b. 
2148     =  4  m. 
551     =  t. 

6)3223 

537-2  =  Prismoidal  Mean  Area. 
And  for  Cubic  Yards,  in  100  feet  long,  per  Table  =  1989'6. 

Example  4. — Although  embankment  is  merely  excavation  inverted, 
and  governed  in  its  computation  by  precisely  the  same  principles,  we 
will  now  give  an  example  of  embankment  on  irregular  or  nine-level 
ground,  road-bed  16,  side-slopes  1J  to  1,  and  ground  surface  supposed 
to  be  jagged  masses  of  rock.  CC  represents  as  usual  the  centre  or 
guiding  line  of  the  road,  the  cross-sections  being  dimensioned  »* 


CHAP.  II.— FIRST   METII.  COUP.— ART.  18. 


89 


Fig.  59 


marked  upon  the  figures  (62,  63,  64),  the  distance  between  the  end 
sections  being  a  regular  station  of  100  feet,  and  m  (Fig.  63)  being  the 
hypothetical  raid-section,  deduced  from  the  two  others,  supposed  to 
have  been  regularly  measured  by  the  field  engineer,  and  furnished  to 
the  computer  by  him  from  his  note  book. 

The  areas  of  the  sections  being  given,  having  been  previously  cal 
culated  in  the  customary  manner. 


Example  4 


Cross-sections  and.  areas. 
b    =602 


m  =  691 
*    =  786 
Length,  100  feet 


h. 


90 


MEASUREMENT   OF  EARTHWORKS. 


Calculations  for  solidity ; 

602  =  b. 
2764  =  4  m. 

786  =  t. 
6)4152 

692  =  Prismoidal  Mean  Area. 
And  for  Cubic  Yards,  in  100  feet  long,  per  Table  =  2562*9. 


Fig.  62. 


.  63. 


17. 


10 


1 


10 


As  has  been  observed  ^before,  b  and  t  are  correlative,  and  either 
might  be  taken  as  base ;   the  calculations  of  quantity  are  'usually 


CHAP.  II.— FIRST   METH.  COMP.— ART.  19.  91 

made  in  the  direction  in  which  the  numbers  run,  or  the  one  nearest 
to  us  of  any  pair  may  be  assumed  as  b,  and  the  other  as  t — it  is  quite 
immaterial  which — but  during  the  pendency  of  the  computation,  to 
which  they  are  subject,  the  special  designation  must  remain  for  the 
time  unchanged. 

The  surface  of  ground,  assumed  in  this  example,  appears  to  be  suf- 
ficiently irregular  to  test  any  rule  (though  rougher  ones  will  occur  to 
the  memory  of  most  engineers),  and  we  might  proceed  to  give  illus- 
trations of  such,  but  enough  has  been  done  in  this  way  to  indicate  the 
principles  on  which  we  work,  and  which  can  readily  be  applied  to 
any  case  which  may  occur  in  practice.  Nor  does  it  seem  necessary 
here  to  define  and  classify  the  numerous  distinct  cases  of  earthwork — 
the  Prismoidal  Formula  holds  for  all,  and  it  is  left  to  the  judgment 
of  the  engineer  to  make  the  application. 

19.  Connected  Calculation  of  Contiguous  Portions  of  Excavation  and 
Embankment,  with  the  Passage  from  one  to  the  other. 

Example  5. — See  Figs.  65  to  71. 

In  Fig.  65,  ABC,  a  portion  of  a  railroad  cut,  road-bed  =  20,  side- 
slopes  1  to  1.  BCD,  a  portion  of  a  railroad^,  road-bed  =  14,  slopes 
1£  to  1.  Grade  points  0  four  in  number,  besides  the  centre. 

In  Figs.  66  to  71,  six  cross-sections,  3  of  excavation  and  3  of 
embankment,  are  shown,  and  all  dimensioned  as  marked.  Fig.  68  is 
the  base  of  the  closing  pyramid  of  excavation  in  the  passage  from 
excavation  and  embankment,  the  vertex  of  which  is  at  the  grade 
point  B.  Fig.  69  is  the  base  of  the  closing  pyramid  of  embankment, 
in  the  passage  from  embankment  to  excavation,  the  vertex  of  which 
is  at  the  grade  point  C. 

The  other  cross-sections  are  those  necessary  to  compute  the  portions 
of  excavation  and  embankment  shown  upon  the  plan,  Fig.  65.  One 
of  them  only  is  at  a  regular  station,  called  station  (10),  Fig.  68,  the 
others  are  all  intermediates,  supposed  to  have  been  required  by  the 
configuration  of  the  ground. 

The  scale  is  20  feet  to  the  inch. 

On  the  centre  line,  the  excavation  shown  is  61  feet  in  length — but 
the  closirlg  pyramid  of  cutting  runs  11  feet  further  to  its  vertex  at  the 
grade  point  B.  While  in  like  manner  the  embankment  is  48  feet 
long  on  the  centre,  and  the  closing  pyramid  of  filling  extends  7  feet 
further  to  its  vertex  at  the  grade  point  C. 

This  over-lapping  of  the  closing  pyramids  is  an  inconvenience,  but 
it  is  sometimes  unavoidable. 


92 


MEASUREMENT  OF  EARTHWORKS. 


Plan 


Cross    jSecs. 


Fig.es. 


Sta: 

9+ SO 


9+75 


CHAF.   11.—  FIRST   METH.  COMP.— ART.  19.  93 

Calculations  for  Solidity. 

Position  of  Cross-sec-         Distances  Cross-section 

tions  upon  the  centre.  apart.  Areas,  etc. 

9  +  50    ...       0    ...      342     =  b.       "] 
9  +  75     ...     25     ...       907     =  4  w.   [ 
10  Reg.  Sta.    .     .     25     .     .     .       106     =  t.         }~  Excavation. 

Length  =  50  f>)1355 

225'8  =  Prism.  Mean  Area. 


418-1  =  Cubic     Yards,     by 

Table  for  /&  feet  =  418'1 
10+11  Grade  at  centre. 

(Paf9aget  etc.,  from  Excavation  to  Embankment.) 
Closing  Pyramid  of  Excavation,  vertex  at  Br  Fig.  6& 
Area  of  base  at  10  =  106.     Theny 

1  C\f  _L  1  Oft    i     f\    Mean  Area. 

tJp     -  =  35-3  Xlengthy22=by  Table  1307  Xr202o=    28'8 
Total  Solidity  of  Excavation =  446*9 

Now,  commence  the  embankment  with  the  closing  pyra- 
mid in  the  passage,  altitude  or  length  15  feet,  and  vertex  at 
C,  Fig.  65.  Area  of  base  at  10  +  19  =  46.  Then, 

AP  _|_  Afi  _l_  0    ^ean  AreR- 

L±L     Z  =  15-3  X  length,  15  =  by  Table  56'7  X  T\ft  =      8*5 

10  +  19    .    .   '.      0    ...      46     =  b. 

10  +  39     .    .-.    20    ...     504     =  4m. 

10  +  59     ...     20     ...     215-5  =  t.  }•  Embankment. 

Length  =40  6)765'5 

127 '6  =  Pris.  Mean  Area. 

189-0  =  Cubic      Yards,      by 

.  Table  for  ffo  =  189'0 

Total  Solidity  of  Embankment =  197'5 

And  this  closes  the  computation  of  Cubic  Yards  in  the  portion  of 
Excavation  and  Embankment,  from  A  to  D  (Fig.  65),  including  the 
passage  between  them,  and  comprising  in  all  two  prismoids  and  two 
closing  pyramids. 

In  concluding  this  branch  of  the  subject,  we  may  mention  that  as 
HUTTON  defines  "  a  pri&moid  "  to  have  in  its  end  sections  "  an  equal 
number  of  sides"  (Arts.  3  and  14),  a  like  number  of  level  hights,  or 


1)4  MEASUREMENT  OF  EARTHWORKS. 

cuttings,  ought  always  to  be  taken  in  adjacent  cross-sections,  but 
should  that  have  been  omitted  in  the  field,  additional  cuttings  may 
be  computed  or  drawn  upon  the  sections  obtained,  so  that  previous 
to  calculating  their  areas,  there  shall  be  the  same  number  of  cuttings  in 
all  the  adjacent  cross-sections,  and  we  shall  then,  have  for  solidity  a  correct 
prismoid. 

a In  verifying  the  work  given  in  the  first  four  examples 

preceding — illustrated  by  Figs.  53  to  63  inclusive — the  end  areas  and 
length  being  correctly  given  in  all,  it  is  only  necessary  to  prove  the 
mid-section  ;  as  an  agreement  there  necessitates  a  like  result  when 
used  with  the  given  d&ta,prismoidally,  to  find  the  solidity. 

This  proof  may  be  made  either  by  our  2d  method  of  computation 
(Rights  and  Widths),  or  3d  method  (Roots  and  Squares) — the  latter 
being  generally  the  most  convenient,  though  the  former  may  often 
be  used  with  advantage. 

No  single  calculation,  truly  says  Prof.  Gillespie,  ought  ever  to 
be  relied  on  by  the  engineer,  and  proof  of  the  correctness  of  every 
computation  should  always  be  obtained  before  employing  it  in  work. 

It  is  often  the  case  when  railroads  follow  the  rugged  margins  of 
rivers  that  many  miles  of  side-hill  work  present  themselves,  where 
the  road-bed,  located  above  the  flood  line,  lays  in  rock  excavation  on 
one  side,  and  heavy  embankment  upon  the  other— to  such  cases  the 
preceding  method  of  computation  will  be  found  peculiarly  applicable ; 
both  cutting  and  filling  showing  themselves. upon  the  end  cross-sec- 
tions of  every  station  and  intermediate,  while  the  mid-section  may  be 
diagrammed  between  them  with  great  facility. 

In  continuing  this  chapter  we  may  state — That  in  any  right-lined 
solid  whatever,  lying  between  two  parallel  planes  (according  to  the 
definition  of  a  prismoid),  whenever  a  mid-section  can  be  correctly 
deduced  between  two  given  end  sections,  situated  in  the  limiting 
planes  (and  by  taking  pains  it  always  can  be),  there,'  our  First  Method 
of  Computation  will  be  found  to  apply  strictly  for  solidity. 

So  that  this  method  is  a  standard  test  for  all  other  rules,  and  has  been 
accepted  as  such  by  Prof.  Gillespie,  and  other  able  writers. 

Hence,  we  may  repeat  that  the  formula  employed  in  this  chapter 
i?  the  fundamental  rule  for  the  mensuration  of  all  right-lined  solids, 
within  parallel  planes,  and  applicable  also  to  many  warped  figures, 
and  other  curvilinear  bodies,  in  a  manner  so  unexpected  as  to  have 
excited  the  surprise  of  some  able  geometers,  whose  attention  had  not 
been  specially  directed  to  that  subject  before. 


CHAP.  II.— FIRST  METH.  COMP.— ART.  19.  95 

Cases  often  occur  in  heavy  work,  where  it  is  evident  from  the  cross- 
sections,  that  the  bulk  of  the  solid  under  consideration  lays  consider- 
ably on  one  side  of  the  centre  line  (or  where,  in  common  phrase,  the 
sections  are  lop-sided),  and  it  would  seem  in  such  cases  as  if  some 
correction  ought  to  be  made  for  the  position  of  the  centres  of  gravity 
(as  indicated  upon  Figs.  43  and  44,  Chapter  I.)  ;  for  it  is  most  obvious 
that  in  a  long  line  of  heavy  work  the  path  of  gravity  centres  would 
frequently  Gross  and  re-cross  the  guiding  line  of  the  work,  and  hence 
would  necessarily  be  longer. 

So  that  if  the  line  of  magnitude  should  be  assumed  as  the  true 
line  of  calculation,  the  centres  of  gravity  ought  to  be  assembled 
upon  the  centre  line,  in  effect,  at  every  station,  and  this  correction 
would  probably  be  found  by  multiplying  the  projections  of  the  points 
of  gravity  upon  the  centre,  by  their  distances  from  it  (-f  when  oil 
the  same  side  —  when  opposite)  ;  but  this  is  a  refinement  which  has 
never  been  employed  by  engineers,  in  dealing  with  the  huge  masses 
in  question. 

What  the  engineer  most  needs  in  earthworks  appears*  to  be — not 
astronomical  accuracy,  but  the  systematic  use  of  some  rule  for  solidity, 
which  shall  always  be  consistent  with  itself,  and  closely  proximate 
the  truth,  without  involving  those  stupendous  discrepancies  (men- 
tioned by  many  writers),  as  flowing  from  the  employment  of  the 
average  methods,  which  have  been  so  much  (and  as  it  always  appeared 
to  the  writer)  so  unnecessarily,  used  in  the  ordinary  computations  of 
earthwork. 

The  method  of  computation  developed  in  this  chapter  finds  appro- 
priate application  also  in  masonry  calculations.  In  this  manner  the 
writer  once  computed  the  contents  of  a  heavy  stone  aqueduct,  con- 
taining over  4000  perches,  with  numerous  projections  and  off-sets,  and 
walls  battered,  both  inside  and  outside. 

The  process  taken  was  by  drawing  to  a  scale  accurate  horizontal 
plans,  at  all  the  off-set  levels,  at  the  skewbacks,  and  other  breaks  in 
the  contour — deducing  mid-sections  between  these,  and  multiplying 
together  each  set  of  three,  in  accordance  with  the  Prismoidal  For- 
mula, etc.  • 

This  gave  a  very  satisfactory  exhibit  of  the  work,  and  a  correct 
result  in  volume,  with  less  labor,  and  greater  accuracy,  than  any  other 
modes  he  found  in  use  at  the  time. 

In  calculating  stone  culverts,  and  bridge  abutments  also,  this 
method  will  be  found  quite  useful. 


96  MEASUREMENT   OF  EARTHWORKS. 

In  fact,  in  computing  the  volume  of  solid  bodies  of  any  kind,  the 
engineer  will  find  the  Prismoidal  Formula  to  be  either  strictly  correct, 
or  a  very  close  approximation. 

b We  now  conclude  this  chapter  by  some  remarks  upon 

Borden's  Problem. 

Some  examples  acquire  celebrity  from  being  apposite  in  themselves, 
for  the  illustration  of  important  processes,  and  are  consequently 
copied  by  others  ;  besides,  there  is  an  evident  advantage  to  the  reader 
in  re-producing  examples,  which,  having  been  before  discussed,  are 
more  generally  known  ;  amongst  such  is  Borderis  Problem,  first  pub- 
lished  by  Simeon  Borden,  C.  E.  (Boston,  1851),  in  his  "  System  of 
Useful  Formulae"  (Art.  63). 

He  treats  this  example  at  great  length  (14  pages),  and  commits 
some  errors,  which  were  subsequently  pointed  out  and  corrected  in 
Henck's  Field  Book  (Boston,  1854). 

This  example  was  also  adopted  by  John  Warner,  A.  M.,  in  hia 
Earthwork  (Philadelphia,  1861,  Art.  112),  without  comment. 

The  problem  appears  to  have  given  Mr.  Borden  some  trouble, 
involving  a  number  of  his  "  blind  pyramids,"  and  also  some  errors,  ag 
Mr.  Henck  hath  shown. 

Nevertheless,  it  is  simply  a  case  of  injudicious  cross-sectioning — for 
had  Borden,  instead  of  attempting  to  compute  its  full  length  of  100 
feet,  imagined  an  intermediate  at  50  feet  (for  which  he  gave  all  the 
data  necessary),  all  difficulty  would  have  vanished,  and  he  would 
neither  have  stumbled  over  his  own  blind  pyramids,  nor  been  shortly 
corrected  by  a  subsequent  author. 

Indeed,  Mr.  Borden  admits,  page  186,  of  his  work  of  1851,  that 
"  the  engineer  would  be  likely  to  divide  the  section  into  two  or  three  " 
— and  this  the  present  writer  deems  to  be  not  only  likely,  but  absolutely 
certain. 

Now,  taking  the  end  areas  alone  (100  feet  apart),  and  disregarding 
(for  the  moment)  the  irregularities  of  the  ground,  which  ought  to 
have  been  intercepted  a*nd  brought  out,  by  an  intermediate  at  50 
feet — we  find  : 

Warner,  in  Art.  112,  of  his  Earthwork,  gives  for 

the   volume »     .     .     .  =  1155'9  C.  Yards. 

By  Hutton's  General  Rule  (as  in  this  chapter)  =  1155'9        " 
Difference  .     .     .     .  =     0 

But  Henck,  in  his  Engineer's  Field  Book  (after  noting  Borden's 
mistake  of  360  cubic  feet),  finds  by  his  own  process  the  solidity  = 


CHAP  II— FIRST  METH.  COMP.— ART.  19.  97 

32,820  cubic  feet  =  1215*5  cubic  yards ;  or,  the  former  are  in  a 
deficiency  of —  59*6  cubic  yards,  an  error  inadmissible  in  the  quan- 
tity before  us. 

In  this  problem  Borden  makes  two  theoretical  suppositions,  and 
two  summations  of  results,  based  upon  his  hypothetical  view  of  the 
effect  upon  solidity  of  the  irregularities  of  the  ground  surface,  between 
the  end  sections,  but  he  gives  no  opinion  on  either. 

The  Prismoidal  Formula  of  Hutton  (computed  on  the  whole  sta- 
tion of  100  feet)  gives  precisely  an  Arithmetical  Mean  between  the  two 
suppositions  of  Borden,  but  is  considerably  in  defect  of  the  true  vol- 
ume as  given  by  Henck's  Formula. 

And  here  we  come  to  the  point  of  the  importance  of  properly  cross 
sectioning  a  solid,  before  we  begin  to  calculate  it; — for  if  we  sketch 
from  Borden's  data  an  intermediate  at  50  feet,  of  which  we  find  the 
area  to  be  335*6 — then  all  difficulties  are  at  once  resolved,  and  we  pro- 
ceed prismoidally  in  a  few  lines  to  reach  a  correct  result,  which  Mr. 
Borden  failed  to  attain  in  fourteen  pages. 

Considered  in  connection  with  an  intermediate  at  50  feet,  Borden's 
Problem  stands  as  follows :  Two  end  areas  =  387  and  240.  One 
intermediate  area  =  335*6.  Now,  deducing  between  these  (by  Bor- 
den's data)  the  hypothetical  mid-sections,  required  by  Button's  Gen- 
eral Rule,  we  find  they  have  areas  of  293*5  and  366*5,  and  working 
prismoidally  with  them  we  quickly  find  the  solidity  of  the  entire  body 
to  be  32,820  cubic  feet,  or  1215*5  cubic  yards — precisely  the  same  as 
Henck  makes  it  by  his  own  formula,  and  as  Borden  would  have  made 
it  had  he  been  aware  of  the  errors  into  which  his  own  "blind pyra- 
mids" led  him. 

As  this  problem  is  a  well-known  one,  and  has  not  a  very  irregular 
appearance  in  Borden's  diagram,  we  think  this  a  suitable  place  to 
urge  upon  all  engineers  the  great  importance  of  judicious  cross-sectioning. 

In  terminating  this  chapter,  we  may  safely  state  that  Button's 
General  Rule,  as  applied  to  earthworks  by  the  methods  detailed 
herein,  is  ONE  WHICH  NEVER  FAILS  WHEN  THE  DATA  is  CORRECT. 
7 


CHAPTER    III. 

SECOND   METHOD   OF  COMPUTATION,  BY   HIGHTS  AND   WIDTHS,  AFTER 
SIMPSON'S  ORIGINAL  RULE. 

20  .......  The  Prismoidal  Formula,  as  originally  demonstrated 

by  Simpson  (1750)  —  see  Art.  2  —  was  evidently  designed  for  the  rect- 
angular prismoid  (Fig.  2)  —  its  end  areas  were  obtained  by  multiply- 
ing together  the  Sights  and  Widths;  and  four  times  its  mid-section 
by  multiplying  the  sum  of  the  Hights  by  the  sum  of  the  Widths. 

To  adapt  it  more  conveniently  to  the  triangular  prismoids  of  Earth- 
works, with  side-slopes  drawn  to  intersect  each  other,  the  original 
formula  of  Simpson  (1750),  reduced  to  the  form  subsequently  enun- 
ciated by  Hutton,  as  a  general  rule  (1770),  is  multiplied  by  2,  on  the 
left  side  only,  changing  its  divisor  at  the  same  time. 

Thus, 


2_ 

~~ 


This  is  the  same  thing,  in  effect,  as  the  original  formula  of  Simp- 
son (when  arranged  for  a  mean  area)  ;  for  if  we  suppose  the  rectan- 
gular prismoid  (Fig.  2)  cut  in  half  by  a  plane  through  the  diagonals 
of  its  end  areas,  FB,  etc.,  so  as  to  convert  it  into  two  triangular  pris- 
moids  (each  with  one  right  angle),  the  Hights  X  Widths  from  the 
right  angle  would  give  double  the  triangular  area  of  each  end,  while 
their  sums,  multiplied  together,  would  equal  8  times  the  triangular 
mid-section,  the  divisor  becoming  6  X  2  =  12. 

*  It  would  evidently  be  a  much  better  notation  for  earthwork  to  adopt  I  inttead  of  h, 
because  the  greatest  extent  of  an  earthwork  solid  usually  lays  along  the  ground  (length- 
wise) ;  but  Simpson  and  Hutton,  the  fathers  of  these  formulas,  have  both  used  h  —  they 
dealing  generally  with  prismoids  of  small  dimensions,  supposed  to  stand  erect  upon  a 
base  (as  in  Figs.  1  and  3),  and  have  been  followed  by  most  writers,  and  necessarily  for 
the  most  part  also  here  ;  though  we  have  occasionally  used  I  (to  avoid  confusion),  and 
this  must  be  taken  as  correllative  with  the  h  of  Simpson  and  Hutton,  in  the  cases  in 
which  it  has  been  employed;  but  some  care  will  be  needed  to  avoid  confounding  the  h 
indicating  the  length  of  the  prismoid,  with  the  same  letter  often  used  as  a  symbol  for 
{light  In  cross  sections. 

98 


CHAP.  III.— SECOND  METH.  COMP.-ART.  20.  99 

Now,  as  shown  in  Art.  8,  a,  it  is  an  equivalent  process  to  imagine 
the  triangular  section,  partially  revolved,  so  as  to  bring  the  edge  of 
the  diedral  angle  downwards,  and  to  cause  its  bisector  (the  centre  line) 
to  become  the  perpendicular  hight  (h)  of  the  cross-section,  while  the 
extreme  breadth  to  ground  edges  of  side-slopes,  horizontally,  becomes 
the  width  (w) — then,  by  Art.  8,*  we  have  h  X  w  =  double  area  of 
triangular  section  to  intersection  of  side-slopes. 

This  is  the  position  occupied  by  the  triangular  areas  of  the  cross- 
sections  of  the  solids  forming  the  earthworks  of  railroads,  the  centre 
line  being  the  bisector,  or  hight  (Ji),  and  the  sum  of  the  distances  out, 
to  the  ground  edges  of  the  side-slopes  of  an  equivalent  triangle,  being 
the  width  (w). 

The  equivalent  triangle  is  often  formed  by  means  of  an  equalizing 
line,  drawn  (for  convenience)  through  the  lowest  side-bight  of  the 
cross-section,  so  as  to  form  a  figure  of  only  three  sides,  exactly  equiva- 
lent in  area  to  the  cross-section  of  earthwork,  which  is  nearly  always 
more  or  less  irregular  on  the  top,  and  frequently  has  numerous  sides 
for  its  ground  line ; — the  side-slopes,  however,  remaining  generally 
uniform  and  even,  from  station  to  station  (see  Fig.  14). 

The  equation  for  Hights  and  Widths  may  often  take  another  form 
(already  mentioned  in  Art.  9),  which,  at  times,  will  be  found  convenient. 

h   =  Hight  at  one  end. 
h'  =       "      "   other  end. 
w  =  Width  at  one  end. 
w'  =       «      "    other  end. 
I    =  Length  of  mass,  usually 
i, HO      denoted    by    (//)    = 
100,  generally. 

7  ,,    t    .   hu!  +  h'  w 

h  w  -f  h'  w'  -f  -        ' 

Then, X  I  =  S. 

o 


Let 


*  In  any  A,  however  situated :— If  one  angle  coincides  with  the  intersection  (or 
origin,)  of  two  rectangular  axes  (such  as  a  Meridian,  and  an  East  and  West  line,  or  centre 
line,  and  base  of  levels),  and  the  co-ordinates  of  the  other  angles  are  known  (as  by  their 
Lat.  and  Dep.,  or  level  bights  and  distances  out) ;  then,  the  area  of  any  such  A  is  easily 
found. 

Thus,  calling  the  first  angle  0,  and  the  others  in  succession  1  and  2. 

(Lat.  of  IX  Dep.  of  2) -(Lat.  of2X  Dep.  of  1) 
We  have,  —  J— L_  J  =  Area  of  A  required. 

But,  in  the  single  case  of  either  rectangular  axis  cutting  the  6,  then,  instead  of  — 
between  the  products  (forming  the  numerator  above)  put  -f .  With  this  exception,  the 


100 


MEASUREMENT  OF  EARTHWORKS. 


This  formula  may  be  briefly  called  (from  a  leading  feature  in  the 
process),  the  direct  and  cross  multiplication  of  Hights  and  WidtJis,  which 

may  be  represented  as  below ;  and  then,  (x  ~),  or  one-sixth  the  whole 
being  taken  =  Solidity. 


Thus, 


For  example,  take  Figs.  72  and  73  (dimensioned  as  marked). 
1.  By  Direct  and  Cross  Multiplication  of  Hights  and  Widths. 

(  h  w  =  23-4  X  47     .     .     .     .  =  1100  Double  area. 
'  \h'w'=  27-6  X  55-5  .    .    .    .  «  1532      " 
and 

Cross 


•oss  Multi-  f  W  =  23-4  X  55'5  =  1299  \ 
plication.   \  h'  w  =  27'6  X  47     =  1297  J 


Let 


h   =  +  23-4 
w  =       47 
h'  -B  +  27-6 
vf  =       55-5 


2)2596  f     Kepresenta- 

1298    =  1298  I  tive     product 
6)3930   I  for  mid-sec. 

( Including  the 

Prism.  Mean  Area  =    655  <      grade  trian. 

of  100  area. 


i 


2.  Proof  by  Simpson's  Formula  (modified  for  triangles). 

Eights.        Widths. 

23-4  X  47  =  1100 
27-6  X  55-5  =  1532 
51  X  102-5  =  5228 


12)7~860 


Prism.  Mean  Area 


655   05    above,   including 
grade  triangle. 


Then,  the  mean  area  X   length  =  100  feet  between  sections 
Solidity  =  65,500  cubic  feet. 


rule  is  general,  and  finds  ready  application  in  computing  the  areas  of  irregular  cross-sec- 
tions, and  the  contents  of  LAND  SURVEYS. 

(Prob.  V.,  Young's  Analyt.  Geom.,  London,  1833.— Prof.  Johnson's  ed.  of  Weisbach, 
Philada.,  1848,  article  107.) 


CHAP.  III.— SECOND  METH.  COMP.— ART.  21.     1Q1 

21 Examples  of  the  Application  of  Simpson's  Rule  to  Earthworks. 

In  further  illustration  of  this  subject,  suppose  Figs.  72,  73,  74,  and 
75,  to  be  cross-sections  upon  a  railroad  line,  in  stations  of  100  feet, 
apart  sections,  with  road-bed  of  20,  side-slopes  1  to  1,  and  other  data 
as  dimensioned  upon  the  figures  given ;  with  equalizing  lines  properly 
drawn,  reducing  them  to  equivalent  triangles,  and  with  centre  hights 
correctly  ascertained. 

Then,  to  find  the  End  Areas  to  Intersection  of  Slopes. 


Hights.            Widths.          Sq.  Ft. 

Fig.  72  =  23-4    X  47     =  1100 
73  =  27-6    X  55-5  =  1532 
74  =  28-8    X  59-9  =  1725 
75  =  27-25  X  54-6  =  1488 

Double  Areas 
in 
Whole  numbers. 

Or,  they  may  be  computed,  as  is  usual  with  engineers,  by  means 
of  trapezoids  and  triangles,  as  they  have  been,  indeed,  in  this  case  for 
the  purpose  of  verification,  and  found  to  agree  in  whole  numbers: 
there  being,  as  usual,  small  differences  in  the  decimal  places. 

When  the  ground  surface  is  irregular,  as  shown  in  these  cross-sec- 
tions, the  successive  processes  are  as  follows: 

1.  Find  the  equalizing  line  by  Art.  8. 

2.  Ascertain  the  centre  hight  from  intersection  of  slopes  to 
equalizing  line. 

3.  Find  the  extreme  width,  or  sum  of  distances  out,  to  the 
edges  of  tops  of  slopes,  where  they  cut  the  equalizing  line. 

4.  Find  the  double  areas  of  the  cross-sections,  by  multiplying 
together  the  hights  and  widths,  or  h  X  w. 

5.  Find  8  times  the  mid-section,  by  means  of  sum  of  Hights  X 
sum  of  Widths. 

6.  Then,  for  Solidity,  proceed  prismoidally,  by  Simpson's  For- 
mula as  modified,  for  triangular  solids. 

The  areas  of  the  cross-sections  haying  been  duly  verified,  we  may 
proceed  to  the  calculation  of  some  examples,  as  follows: 


102 


MEASUREMENT  OF  EARTHWORKS. 


CHAP.  III.— SECOND   METH.  COMP.— ART.  21.  1Q3 

EXAMPLES. 
Figs.  72  and  73. 

(Hights.  Widths. 

23-4  X    47     =  1100      =  Double  Area  of  top. 
27-6  X    55-5  —  1532      =       "  "        base. 

51      X  102-5  =  5228      =  8  times  mid-section. 
12)7860 

655      =  Prismoidal  Mean  Area. 
100       Distance  apart  sections. 
65500  =  Solidity  in  Cubic  Feet. 

Figs.  73  and  74. 

Rights.  Widths. 

27-6  X    55-5  =  1532      =  2  t. 
28-8  X    59-9  =  1725      =26. 
56-4  X  115-4  =  6509      =  8  m. 
12)9766 

814      =  Prismoidal  Mean. 
100 

SHOO  =  Solidity. 
Figs.  74  and  75. 


Hights. 

28-8 
27-25 

X 
X 

Widths. 

59-9  = 
54-6  = 

1125 
1488 
6418 

=  26. 
=  8m. 

56-05 

X 

114-5 

= 

12)9631 

803 
100 

=  Prismoidal  Mean. 

80300  =  Solidity. 

(Cub.  Ft.  ~\       Grade  Prism  to  be  deducted, 
SU  «nd  *«  voiume^ftomnmd. 
ftHQnn  I  bed  to  ground. 
oUoUU  J 
227200  =  Sum  of  quantities. 


Grade  Prism. 


(Then,  227,200  —  30,000  =.       ~—  =  7304  Cubic  Yards. 

Tabulated  by  our  3d  Method  of  Computation  (Roots  and  Squares), 
the  sum  of  the  quantities,  from  Fig.  72  to  Fig.  75  =  227,170  Cubic 
Feet  (including  Grade  Prism)  ;  the  slight  difference  of  30  Cubic  Feet 


104  MEASUREMENT  OF  EARTHWORKS. 

arising  from  neglect  of  decimals  on  both  sides  ; — had  these  been  car- 
ried further,  the  results  would  probably  have  been  identical,  or  very 
nearly  so. 

We  may  also  verify  this  calculation  by  means  of  multipliers, 
modelled  after  Simpson's,  and  applied  to  the  areas,  as  given  in  the 
examples,  as  follows: 

Cross-sections  figured  in  Nos.  72,  73,  74,  and  75,  stations  100  feet. 

Double 
Sta.  Areas,  etc.      Mults.  Sq.  Ft. 

72  1100  X  0-5  =  550 
8  times  mid-sec.  5228  X  0'5  =  2615 

73  1532  X  1     =  1532 
8  times  mid-sec.  6509  X  0'5  =  3255 

74  1725  X  1     =  1725 
8  times  mid-sec.  6418  X  0'5  =  3209 

75  1488  X  0-5  =  744 

6)13630 
2272 

100  Double  Interval. 
Solidity,  in  Cubic  Feet  =    227,200,  same  as  before. 

The  intervals  are  subdivided  by  the  mid-sections  into  50  feet 
epaces,  or  single  interval.  The  regular  stations  of  100  feet  forming  a 
double  interval  in  this  case. 

The  Grade  Prism  being  deducted  (30,000  Cubic  Feet),  and  the 
remainder  divided  by  27,  we  have  as  before,  a  volume  of  7304  Cubic 
Yards. 


22.  Observations  upon  Simpson's  Rule.  SIMPSON  appears  to  have 
framed  his  rule  for  application  to  rectangular  prismoids,  and  as  such 
he  demonstrated  it  in  reference  to  a  diagram  like  Fig.  2,  Art.  2 — 
including  of  course  those  right  triangles  which  are  the  halves  of 
rectangles. 

He  could  have  had  no  conception  of  the  vast  masses  of  earthwork 
needed  upon  the  public  works  of  later  days  ;  nor  of  providing  a  rule 
for  the  mensuration  of  such  ;  nor,  indeed,  of  the  immense  range  the 
Prismoidal  Formula  has  since  taken. 

His  rule  (see  Art.  2),  though  wonderfully  flexible  when  applied  to 
rectangular  or  triangular  figures,  has  no  leading  lines,  common  with 


CHAP.  III.— SECOND  METH.  COMR— ART.  22.  1Q5 

irregular  ground;  such  surfaces  then  require  to  be  equalized,  by  a 
single  line  on  the  principle  of  Fig.  14* — converting  the  sections 
bounded  by  them  into  equivalent  triangles  before  they  can  be  com- 
puted by  the  Rights  and  Widths  of  Simpson's  Rule,  though  we  find 
occasionally  that  trapezium  sections  also,  when  not  very  much  dis- 
torted, are  often  computable  by  the  rule  mentioned. 

But,  in  applying  such  a  rule  to  the  rude  masses  of  earthwork,  so 
common  at  the  present  day,  failing  cases  were  to  be  expected,  and  the 
peculiar  solid  shown  in  Figs.  81  and  82  furnishes  an  example  in  point. 

Figs.  81  and  82,  Chap.  V.,  computed  by  Simpson's  Rule. 

Eights.      Widths. 

60  X    40  =    2400 


30  X    60  =    1800 


90  X  100  =    9000 
12)13200" 

Prism.  Mean  Area  =    11QO 
Common  length     .  =          100 

Solidity    .    .    .     .  =   110,000  Cubic  Feet. 


But,  by  various 
examples,  in  Arts 
29  and  30,  Chap. 
V.,  the  Solidity  = 
130,000  Cubic  Feet. 


So  that,  in  the  case  of  this  peculiar  solid,  Figs.  81  and  82,  Simp- 
eon's  Rule  falls  short  =  20,000  Cubic  Feet. 

As  the  solid  referred  to  has  one  end  section  a  Rhomboid — the  mid- 
section  a  Pentagon — and  the  other  end  a  Triangle. 

We  could  hardly  expect  Simpson's  Rule,  framed  for  rectangular  and 
triangular  sections,  to  answer  in  a  case  like  this,  and  hence  we  men- 
tion it  especially. 

For  all  the  solids  which  present  sections,  such  as  Simpson  con- 
templated, his  rule  is  unquestionably  correct,  while  it  is  remarkably 
plain  and  simple  in  its  application. 

Further  to  illustrate  what  may  be  expected  from  Simpson's  Rule, 
when  applied  by  equalizing  lines  to  rough  and  heavy  sections,  we  will 
now  compute  the  cases  shown  by  Figs.  43  and  44,  Chapter  I. 

Example,  Illustrated  by  Fig.  43,  Chapter  L 

Side-slopes  1  to  1.  No  road-bed  designated.  Proximate  Computa- 
tion, by  Simpson's  Rule,  to  intersection  of  slopes ;  other  dimensions  as 
in  Fig.  43. 

Equalizing  line  of  base  =  b  =  14°    2'  asc. 
top    =  t  =  15°  57'  asc. 

*  In  substance,  this  method  is  found  in  Button's  Land  Surveying  (1770),  quarto  Mens. 


106 


MEASUREMENT  OF  EARTHWORKS. 


Both  these  lines  being  drawn  from  the  lowest  side-hight,  so  as  to 
equalize  the  areas,  as  per  Fig.  14,  Chapter  I. 

Hights.        Widths. 

f  1500  =  6.  i  =  37.5  x    80  =      3000 

Areas  <    720  =  t.  t  =  25-7  X    56  =      1440 

(Length,  100  feet.  63'2  X  136  =      8595*2 


Prism  Mean  Area  =      1086*3 

Length     ....==  100 

Solidity  .  .  .  .  =  108630 
Same,  by  BUTTON  =  108667 
Difference  .  .  .  =  —  37 

Example,  Illustrated  by  Fig.  44,  Chapter  I. 

Side-slope  Is  to  1.  No  road-bed  designated.  Proximate  Computa- 
tion, by  Simpson's  Rule,  to  intersection  of  slopes,  other  dimensions  as 
in  Fig.  44. 

Equalizing  line  of  the  base  b  =  4°  30'  asc. 
top    t  =  1°    5'  des. 

Both  these  lines  being  drawn  from  the 
lowest  side-hight,  so  as  to  equalize  the  areas, 
as  per  Fig.  14,  Chapter  I. 

Highta.  Widths. 

22-02  X    66     =      1453 
29-81  X    90-7  =      2704 
8122 


f  1352  =  6. 
Areas  <    726=  t.  % 

(Length,  100ft. 


51-83  X  156-7  = 


12)  12279 


Prismoidal    Mean   Area  =      1023'25 

Length =  100 

Solidity =  102325 

By  Wedge  and  Pyramid  =  102363 
Difference    .     .     .     .     .  =  ~—  38 
With  several  other  methods,  this  proximate  calculation  agrees  within 
a  few  cubic  yards. 

Example  from  Warner's  Earthwork,  Art.  86. 
A  heavy  embankment.     For  details,  see  Chapter  V.,  near  the  close. 

(2411  =  b. 
907  =  t. 
Length,  100  feet 
Surface  slope,  15°. 


CHAP  III.— SECOND  METH.  COMP.— ART.  22.  107 

Hignts.         Widths. 

36-7  X  131-4  =    4822 

22-5  X    80-6  =    1814 

59-2  X  212-0  =  12550 
12)19186 
Prismoidal  Mean  "Area   .     .  =    1599 

Length =          100 

Solidity =    159900  Cubic  Feet. 

For  Cubic  Yards  -5-  27  .     .  =        5922 
Deduct  vol.  of  Grade  Prism  =          356 

Solidity =        5566  Cubic  Yards. 

By  Hutton's  Rule .     .     .     .  =        5566 
Difference =          ±0 

In  calculating  by  Simpson's  Rule,  the  example  figured  by  Figs.  74 
and  75 — which  agrees  very  nearly  with  BUTTON — we  observe,  by 
reference  to  the  figures,  that  the  ground  slope  at  the  end  sections 
differs  about  9°.  So  that  we  may  safely  assume  that  where  the 
equalizing  lines  (representing  the  ground)  have  a  nearly  similar 
slope,  and  in  the  same  direction,  which  do  not  differ  more  than  10°  in 
their  inclination,  SIMPSON'S  Rule  may  be  safely  used — this  appears  to 
be  a  sure  limit,  and  we  might  perhaps  go  higher. 

When  the  work  happens  to  be  upon  uniform  ground,  or  the  equal- 
izing lines  have  the  same  slope,  as  in  the  case  cited  from  Warner's 
Earthwork,  where  the  ground  slope  itself  is  uniform  at  15°,  the 
results  obtained  by  Simpson's  Rule  ought  to  be  exact^andf  they  appear 
to  be  so. 


LIBKAU  5f 

VERSITY  OF 

CALIFORNIA. 


CHAPTER   IV. 

THIRD  METHOD  OF  COMPUTATION,  BY  MEANS  OF  ROOTS  AND  SQUARES  J 
A  PECULIAR  MODIFICATION  OF  THE  PRISMOIDAL  FORMULA,  WHICH 
WILL  BE  FOUND  IN  PRACTICE  TO  BE  BOTH  EXPEDITIOUS  A!N  D 
CORRECT,  IN  ORDINARY  CASES. 

23 This  method  of  computation,  by  Roots  and  Squares,* 

appears  to  be  the  most  rapid  and  compendious  one  treated  by  us, 
while  it  requires  less  data  and  preliminary  work,  and  agrees  in  its 
results  (for  usual  field  work)  with  computations  made  direct  by  the 
Prismoidal  Formula,  of  which,  indeed,  it  is  only  a  special  modification, 
more  concise  and  rapid  in  use,  but  at  the  same  time  less  accurate. 
The  formula  for  the  Rule  of  Roots  and  Squares  has  been  already 
described  in  the  Preliminary  Problems,  Art.  10,  where  it  is  num- 
bered XI.,  and  is  as  follows: 


-g-          -  X  /  =  S. 

Where, 

h*  =  Representative  square  of  area  of  top, 

from  ground  to  intersection  of  slopes  =  (f). 
h-2  =  Representative  square  of  area  of  base, 

from  ground  to  intersection  of  slopes  =  (b). 
(h  -f  h'y  =  Representative  square  of  4  times  mid-sec.  =  (4m). 
I  =  Distance  apart  sections — usually  desig- 
nated as  (h)    by  the  earlier  writers, 
and   hence  continued  by  us  to  some 
extent ;   though  I  is  clearly  a  more 
suitable  symbol  for  earthwork,  which, 
with  a  comparatively  small  cross-sec- 
tion,  extends    its    length   along    the 
ground. 


*  This  method  is  materially  aided  in  its  use  by  a  good  Table  of  Squares  and  Roots.- 
Prof.   De   Morgan's   stereotyped   edition  of  Barlow's  Tables    (8vo,    London,   1860)  is 
believed  to  be  the  best: — a  very  large  edition  was  published,  and  this  valuable  work  can 
be  obtained  from  any  of  our  importing  booksellers  at  quite  a  low  price. 

When  the  numbers  are  large,  the  well  known  method  of  Logarithms  gives  the  simplest 
process  for  Involution  or  Evolution. 

108 


CHAP.  IV.—  THIRD  METH.  COMP.—  ART.  23.  109 

Note.  —  That  the  bights  of  the  end  sections  in  this  chapter  are 
always  to  be  considered  as  extending  from  the  ground  to  intersection 
of  slopes,  or  be  representative  of  such. 

The  most  important  item  in  this  notation  is  (h  -f  /i')2,  which,  by 
geometry,  we  know  to  be  equivalent  to  4  ^  —  -  —  j  ,  while  —  -  — 
is  the  representative  in  the  mid-section  of  a  line  similar  to  h  and  h'. 

So  that  this  formula  (for  a  single  station)  is,  in  fact,  equivalent  to 
the  Prismoidal  Formula,  as  heretofore  expressed,  viz.  : 


but  for  exact  work  (our  formula  above)  requires  the  end  sections  to 
be  triangles,  with  a  uniform  ground  slope. 

Let  us  now  apply  the  above  formula  to  an  entire  cut  or  bank,  to 
be  computed  by  Mutton's  Kule  (adopted  from  Simpson)  —  see  Art.  10, 
Formula  IX. 

Where  A+4B-f  20  ^  I)vM6  interval  =  S. 

D 

Here,  for  a  case  of  6  single  or  3  double  intervals,  as  shown  —  in  the 
skeleton  table  —  below. 

We  have,  for  3  double  intervals  or  even  spaces  between  stations  of 
equal  length  : 

A2  -f  7t'2  .     .     .  =  A.  The  sum   of  extreme  sections,  each  desig- 

nating one  end. 

3  (h  -f  h')2  .     .  =  4  B.  Mid-sections,  standing  on  even  numbers. 
2  (h'y  +  2  (7i)2  =  2  C.  Kegular  Cross-sections,  standing  on   odd 

numbers. 

Double  Interval  —  Any  one  of  the  uniform  spaces,  from  1  to  3,  or 
3  to  5,  etc.,  being  the  odd  numbers  where  the  regular  cross-sec- 
tions stand. 
S  =  Solidity  of  entire  cut  of  3  equal  stations  in  length. 

Example  1  ......  Being  a  simple  case  (on  irregular  ground)  of 

three  uniform  stations,  or  double  intervals,  of  100  feet  each,  the  mid- 
sections  falling  in  between,  and  dividing  the  length  of  300  feet  into 
single  intervals  of  50  feet  each  ;  for  which  we  will  tabulate  the  exam- 
ple represented  by  Figs.  72,  73,  74,  and  75,  of  Chapter  III.  —  in  a 
skeleton  table  —  as  follows  : 


110 


MEASUREMENT  OF  EARTHWORKS. 


/«» 

(/>+><')' 

1,'* 

(h+h')* 

/ia 

(h+h1)* 

h» 

1 

3 

5 

7 

Regular  stations  designated  by 
the  numbers  of  the  figures. 

72 

73 

74 

75 

Places  of  mid-sections,  on  even 
numbers. 

2 

4 

6 

Regular  cross-section  areas,  upon 
the  odd  numbers. 

550- 

766- 

862-5 

744- 

Square  roots  of  areas  ot  regular 
cross-sections. 

23-45 

27-68 

29-37 

27-28 

Sums  of  square  roots. 

51-13 

57-05 

5665 

*  Squares  of  sums,  or  4  times  the 
proper  mid-section. 

2615- 

3255- 

3209- 

Extra 
decimals 
thrown 
together 
here. 

Having  given  the  skeleton  table  of  data,  we  will  now  tabulate  for 
solidity  on  three  different  plans,  any  one  of  which  may  be  adopted,  01 
in  fact  any  other  which  truly  represents  the  formula  given. 

Tabulation  for  Solidify. 


On  the  plan  of  Art.  10,  iu  Chap- 
ter I. 

By  Simpson's  Rule  (as  given  by 
Hntton). 

By  Multipliers,  modelled  after 
Simpson's. 

Sta.  72.    Areas  .     .  =        550 
4micJ-8ec.    .    .          2615 

73.      .  //    ;«j 

A.          4  B.        2  C. 

550         2615        766 
744         3255        766 

End  areas, 
and  4  times 
mid-section.  Mults.  Results 

4  mid-sec.    .    .          3255 
7                            f        862-5 

1294         3209        862-5 
4  B  =    9079        862-5 

2615     X  1  =    2615 
766     X  2  =    1532 

4  mid-sec.    .    .          3209 
75  744 

2C  =-    3257        3257 
A     =    1294 

3255     X  1  =    3255 
862-5  X  2  =    1725 
3209     X  1  —    3209 

6)13630 

6)13630 

744     X  1  =      744 

General  Mean  Area  =      2271-7 
Double  Interval.    .  =           100 

2271-7 
100  Double  Int. 

6)13630 
2271-7 

Solidity  in  C.  Feet  =     227.170 

Solidity  =    227,170  in  C.  Feet. 

Double  Interval   .    .  =         100 

Whole  length  of  cut  300  feet. 

Whole  length  of  cut  300  feet. 

Solidity   in    C.  Feet  =  227,170 
Whole  length  of  cut  300  feet. 

24.  Now,  for  further  illustration  :— Take  any  cut  or  bank — say  of 
6  (or  any  even  number  of)  equal  stations — their  termini  being  tem- 

*  HtiTTON  and  other  geometers  have  shown  that  the  square  of  any  line  equals  4  times 
that  of  half  the  line ; — and  that  similar  triangles  are  to  each  other  not  only  as  the  squares 
of  their  like  sides,  but  also  as  the  squares  of  any  similar  lines;  and  these  principles  of 
Geometry  lay  at  the  foundation  of  the  method  of  computation,  developed  in  this  Chap- 
ter IV.  (as  already  indicated  in  the  Preliminary  Problems). 


CHAP.  IV.— THIRD  METH.  COMP.— ART.  24. 


Ill 


porarily  numbered  in  the  series  of  odd  numbers,  while  the  interme- 
diate spaces  (or  places  of  mid-sections)  are  also  temporarily  numbered 
in  the  series  of  even  numbers,  and  the  places  of  cross-sections  and  mid- 
sections,  as  well  as  those  of  the  symbols  used  in  the  formula,  all 
regularly  marked,  as  follows : 


Regular  stations. 
Places  of  cross-sees. 
"      mid-sees. 
Symbols  of  formula 


10      6    1 

(h+h'f\  h*    (/i+A')3| 


10 


19 


This  little  skeleton  table  shows  the  positions  of  the  representative 
squares  equivalent  to  the  areas  of  the  several  regular  cross-sections 
computed,  and  also  of  4  times  the  proper  mid-sections,  which  belong 
between  them,  and  it  will  indicate  the  manner  in  which  they  are 
combined  relatively  to  the  odd  numbers,  which  represent  the  regular 
stations ;  so  that  having  computed  the  regular  cross-sections,  we  can 
readily  assemble  them  in  a  skeleton  table,  compute  from  them  by 
Roots  and  Squares  the  other  data  demanded  by  the  formula,  and 
proceed  to  tabulate  for  Solidity,  as  has  been  already  shown,  and  will 
be  more  conspicuously  exhibited  hereafter. 

JJpon  the  foregoing  principles  we  will  now  proceed  with  an  entire 
piece  of  heavy  embankment,  succeeded  by  a  rock  cut,  as  shown  in 
the  annexed,  Fig.  76. 


Example  2.     ...  BANK  =  1000   feet   long.     .    .    .    Fig.   76. 

Skeleton  Table  of  Data,  Given  or  Computed. 

Length  of  regular  ftationi  100  feet— intervals  produced  by  Mid-sections  50  feet. 

Regular  stations  of  100  feet  =    1  2             3             45              6              7              8              9             10        11 

Temporary  numbers    .    .    .  =    1  3             5             7           9             11             13             15             17             19        21 

Regular  Cross-section  Areas  =  24  185         495       1467      3123         3123         3123         1978         1197           391        24 
Places  of  mid-sees.,  inter-  \ 
mediates  at  50  ft. (really).  J 

the  Cross-sec-) 


11' 


|| 


4-90       13-60       22-25       38-30     55'88        55'88 


44-47         34-60        19-77     4-90 


tion  Areas /  " 

gums  of  Roots =      18-50       35-85       60-55       94-18       111-76       111-76       100-35       79-07       64-37      24-67 


Squares  of  Sums,  or  4  time 
the  Mid-section  Areas. 


342-25    1285-22    3666-30    8869-87    12490-30    12490-30    10070-12    6252-06    2956-10    608-61 


*  For  Figs.  77  and  78,  illustrating  a  supposed  basis  of  the  Prismoidal  Formula,  and 
its  connexion  with  Simpson's  Rule  for  Cubature  (see  Chap.  VII.). 


112 


MEASUREMENT  OF  EARTHWORKS. 


CHAP.  IV— THIRD  METH.  COMP.— ART.  24. 

Tabulations  for  Solidity ; 


113 


1. 

Regular  stations 
of  100  feet. 
1    ... 

Cross-sect  iou 
Areas. 
.     .—             24 

2.  By  M! 

Mnl 

1 

2 
1 
2 
1 
2 
1 
2 
1 
2 
1 
2 
1 
2 
1 
2 
1 
2 
1 
1 
Proof 
Gen.mea 

Solidity 

4  times  mid-section 
2   ... 

.     .  =          342-25 
_  f       185 

3   ... 

'     '        (185 
.     .=        1285-82 
_  f       495 

it                   a 
4 

I       495 
.     .  =        3666-30 
f     1467 

€t                         « 

5   .     . 

(     1467 
.     .  =        8869-87 
_f     3123 

6   ... 

*     '       {     3123 
.    .  =      12490 
_f     3123 

7   ... 

'     "       {     3123 
.     .  =      12490 
_  f     3123 

U                           « 

8   ... 

.     .  =      10070-12 
_  {     1978 

9        .     . 

.     .  =        6252-06 

10        .     . 

'     '       {     1197 
.     .  =        2956-10 
„  f       391 

'     *        (       391 

.     .  =          608-61 
—            24 

6^89243-13 

Gen.mean  area  to  int.of  slopes  =  14874 
100 

Solidity  in  c.ft.to  int.of  slopet  —  1487400  of 
BANK. 

By  100  feet  stations,  or  bOfeet  intervals. 

2.  By  Multipliers,  modelled  after  Simpson's. 
Results. 

=         2t 

=       342 

=      370 

=    1285 

=      990 

=    3667 

=    2934 

=    8870 

=     6246 

=  12490 

=     6246 

.  =  12490 


=  10070 

=     3956 

.     .     .     .     .==     6252 

=     2394 

=     2956 

=      782 

=      609 

=        24 

6)89243 
Gen.mean  area  to  int.of  slopes  =  14874 

100 

Solidity  in  c.ft.  to  int.of  elope*  =  1487400  of 

BANK. 


.  Fig.  76. 


Example  2  —  Continued.         ROCK  Cur  =  1000  feet  long.  . 
Skeleton  Table  of  Data,  Given  or  Computed. 

Length  of  regular  stations  100  feet  ;  which,  by  means  of  the  Hypothetical  M  id-  sections,  cover  the  ground  with  50 

feet  intervals. 

Regular  stations  of  100  feet  =     11          12          13          14          15          16          17          18          19          20          21 
Temporary  numbers    .    .    .=       1  3  5  7  9          11          13          15          17          19          21 

Regular  Cross-section  Areas  =,    192         386         646         801         975         768         589         706         771         433         192 
Places  of  mid-sees.,  Inter-) 

mediates  at  50  ft.(really).  j  = 
V  Roots  of  the  Cross-section  ) 

Areas    ......    -J: 

Sums  of  Roots  .....     = 

Squares  of  Sums,  or  4  times  1 

the  Mid-section  Areas.     } 
8 


10 


H 


13-86      19-65      25-42      28-31      31-23      27-71      24-27 

33-51       45-07       53-73       59-54       58-96       51-98 
1122-92    2031-30    2886-91    3545'01    3476'28    2701-92    2584-70    2952-83    2360-01    1202-01 


26-57 
50-84 


27'77 
54-34 


20-81      13-86 
48-58       34-87 


114 


MEASUREMENT  OF  EARTHWORKS. 


Tabulations  for  Solidity : 


By  100  feet  stations,  or  50  feet  intervals. 


1. 

Regular  stations  -~- 

of  100  feet. 

n  .   .   .   .   . 

4  times  mid-section  .     .     . 


Gen.mean  area  to  int.of  slopes 
Solidity  in  c.ft.to  int.  of  slopes 


Cross-section 

2.  By  Mi 

Mul 

Areas. 

1 

:           192 

1 

1122-92 

2 

f       386 

1 

1       386 

2 

2031-30 

1 

f       646 

{       646 

2 

2886-91 

1 

f       801 
{       801 

2 
1 

3545-01 

2 

f       975 

1 

i       975 

2 

:        3476-28 

1 

f       768 

2 

1       768 

1 

:        2701-92 

2 

f       589 
1       589 

1 

2 

2584-70 

1 

f       706 

I       706 

1 

2952-83 

Proof 

f       771 

Gen.mea 

•\       771 

2360-01 

Solidity 

f       433 

'{       433 

1202-01 

192 

6)37397-89 

=   6233 

100 

=   623300  of 

ROCK  CUT. 

2.  By  Multipliers,  modelled  after  Simpson's 
Results. 


=  192 

.  .  .  .  .  =  1123 

....  .  .  =  772 

1 =  2031 

.  .  .  .  .  =  1292 

1 =  2887 

2 =  1602 

.     ...     .=  3545 

...     .     .=  1950 

.     .     ..    .     .=  3476 

=  1536 

1     .......=  2702 

=  1178 

1 =  2585 

=  1412 

.  .  .  . '  .  =  2953 
=  1542 

1 =  2360 

.....=  866 

.  .  .  .  '.==  1202 

.  =  192 


6)37398 
Gen.mean  area  to  int.of  slopes  =   6233 

100 

Solidity  in  c.ft.  to  int.of  slopes  =    623300  of 
KOCK  CUT. 


25.  In  the  preceding  example,  the  side-slopes  of  the  BANK  are  1 J 
to  1  —  road-bed  =  12 ;  while  in  the  ROCK  CUT,  the  side-slopes  are  i 
to  1  —  road-bed  =  16 ;  and  in  all  these  calculations  (we  repeat),  the 
sectional  areas,  in  every  case,  are  taken  from  ground  line  to  intersection 
of  side-slopes ;  and  the  hights,  from  the  vertex  of  the  common  angle 
thus  formed  to  the  line,  or  lines,  representing  the  surface  of  the  ground. 


CHAP.  IV.—  THIRD  METH.  COMR—  ART.  25.  H5 

So  that  in  all  such  computations  —  if  the  contents  above  or  below 
a  given  road-bed  be  desired  in  the  results,  then  the  volume  of  the 
grade  prism  (being  included  in  the  summation)  must  in  every  case 
be  duly  deducted. 

The  volume  of  the  grade  prism  depends  upon  its  sectional  area, 
and  the  length  of  the  bank  or  cut  —  these  calculations  are  very  simple, 
and  once  made,  remain  unchanged  as  long  as  the  road-bed  and  side- 
slopes  continue  uniform. 

Geometers  having  shown  that  the  areas  of  similar  triangles  are  to 
each  other,  not  only  as  the  squares  of  like  sides,  but  also  as  the 
squares  of  any  similar  lines  in  each,  and  these  often  occurring  in 
earthwork  solids,  when  their  cross-sections  are  converted  into  trian- 
gular areas,  by  the  prolongation  (to  a  junction)  of  the  side-slopes,  it 
becomes  of  importance  to  classify  the  relations  existing  among  lines 
and  their  squares,  as  well  as  the  squares  and  rectangles  of  their  sums 
and  differences  ;  —  this  has  been  well  done  in  J.  R.  Young's  Geometry 
(London,  1827),  in  several  successive  propositions  :  —  Book  II.,  4,  5, 
6,  7,  and  8. 

Now,  suppose  any  line  to  be  divided  into  two  parts,  h  and  h'  —  then, 
by  these  propositions,  we  have  : 

1.  (h  +  hj  =  2  (h  +  h')  X 


2.  (h  +  lij  =  A2  -{-  h*  +  2h  h'. 

3.  (h  —  hj  =  h2  -f  h»  —  2/i  h'. 

4.  7i2  —  h"2  =  (h  -f  h')  X  (/*  —  h'). 

5.  7i2  -f  h'*  =  l(h  +  h'y  -f  l(h  —  /O2- 

6.  2  (/i2  -f  7i'2)  =  (h  +  hj  +  (h  —  h')\ 

As  these  lines,  or  parts  of  lines,  may,  and  often  do,  occupy  in  simi- 
lar triangles  the  relation  of  like  lines,  they  become  of  some  conse- 
quence in  earthwork  calculations,  and  in  various  forms  can  be 
traced  through  many  of  the  formulas  now  before  the  public. 

We  will  now  give  an  example  from  Warner's  Earthwork  (Art. 
124),  to  show  that  small  variances  may  be  expected  in  employing  the 
Rule  of  this  Chapter  upon  irregular  ground  :  —  indeed,  it  is  only  in 
uniform  sections  that  an  exact  agreement  of  Rules  can  be  antici- 
pated, but  the  variations  (always  small)  are  not  unlikely  to  balance 
themselves  in  computing  considerable  lengths  of  line. 


116  MEASUREMENT   OF  EARTHWORKS. 

C  End  areas  to  grade  .     .     .     .  =    846'5    .     .  =    915.5 
J  Grade  Triangle  to  add  .     .     .  =    196       .     .  =    196 

(  End  areas  to  int.  of  slopes  .     .  =  1042'5    .     .  =  IIH'5 

H&r6)        Square  Roots =      32'29  .     .  =      33'34 

Sums  of  Roots.    .    ,>,V^H^    .    .  =      65'63 
Square  of  sum,  or 

quadruple  mid-section =  4308 

\  Length,  100  feet. 

Then,  Prismoidally, 

Sum  end  areas  ...'....  =  2154 
Quadruple  Mid-section  .     .     .     .  =  4308 

6)6462 
1077 
Length =         100 

107700 

Off  Grade  Prism =    19600 

27)88100" 
Solidity  in  Cubic  Yards  .     .     .     .  =      3263 

As  computed  by  Warner  (3274,  C.  Y.)  ;  and  also  by  Button's 
General  Rule  (3274,  C.  Y.),  the  difference  made  by  our  Rule  of  this 
Chapter  is,  11  Cubic  Yards,  or  about  $  of  one  per  cent. 

Comparison  of  the  method  of  this  Chapter  with  the  test  examples 
of  Chapter  II.,  as  computed  by  Button's  General  Rule  (each  for  100 
feet  in  length). 

1.   Three-level  Ground. 

(See  Figs.  53,  54,  and  55.)  c.  Yards. 

Computed  by  Roots  and  Squares  (method  of  this  Chapter)  =  2337'6 
"     Button's  General  Rule  (Chapter  II.)  .     .     .  =  2339.6 

Difference =  —  2 

2.  Five-level  Ground. 
(See  Figs.  56,  57,  and  58.) 

Computed  by  Roots  and  Sqirares  (this  Chapter)    .     .     .     .  = 
"        "     Button's  General  Rule  (Chapter  II.)  .     .     .  = 

Difference = 


CHAP.  IV.— THIRD  METH.  COMP— ART.  25.  117 

3.  Seven-level  Ground.  c.  Yards 
Computed  by  Roots  and  Squares  (this  Chapter)    .     .     .     .  =  1990' 

"         "     Button's  General  Rule  (Chapter  II.)  .     .     .  =  1989'6 

Difference =   -f-  0'4 

4.  Nine-level  Ground.  c.  Yards. 
Computed  by  Roots  and  Squares  (this  Chapter)    .     .     .     .  =  2562*9 

"     Button's  General  Rule  (Chapter  II.)  .     .     .  =  2562'9 

Difference =     0 

We  will  now  give  another  example  from  Warner's  Earthwork, 
computed  by  the  method  of  this  chapter. 

Heavy  Embankment  (Art.  86). 

Areas =  2411  907 

\/Roots~    .    .    .    .  =      49-10  30-12 

Sums  of  Roots =      79'22 

Square  of  sum,  ^ 

or  quadruple  >....=  6276 

mid-section.   ) 

Then,  Prismoidally, 

Sum  of  ends  .     .     .  =  3318 
Quadruple  Mid-sec.  =  6276 
6)9594 


X  length  .     .     .     .  =  159900 

-T-  27  for  C.  Yards  =      5566  =  Same  as  Hutton  s  Gen.  Rule. 

From  the  above  it  will  be  observed  that,  with  a  Table  of  Powers 
and  Roots  at  hand,  the  method  of  this  chapter  affords  a  very  convenient 
and  speedy  test  for  volumes,  found  by  other  processes,  and  it  is  a  proxi- 
mately  correct  one. 


CHAPTER  V. 

FOURTH  METHOD  OP  COMPUTATION,  BY  REGARDING  THE  PRISMOID  AS 
BEING  COMPOSED  OP  A  PRISM  WITH  A  WEDGE  SUPERPOSED,  OR  OF  A 
WEDGE  AND  PYRAMID  COMBINED. 

26  .......  Sir  John  Macneill  (1833)  hath  shown  that  a  Prismoid 

of  Earthwork  is  really  a  prism  with  a  wedge  superposed  (as  we  have 
already  mentioned  in  Art.  4)  —  that  the  wedge  is  also  divisible  into 
two  pyramids  —  and  that  the  formulas  for  volume,  in  these  three 
chief  bodies  of  solid  geometry,  form,  by  addition,  the  Prismoidal 
Formula. 

Regarding  the  Prismoid  in  this  way,  and  assuming  it  to  have  been 
diagrammed  as  shown  in  Fig.  8,  Art.  6  (both  end  sections  upon  one 
drawing),  it  is  easily  computable  when  reduced  to  a  level  on  the  top, 
and  the  back  of  the  wedge  is  a  trapezoid,  by  means  of  Formula  VI., 
Art.  6. 

This  Formula  is  : 


-      +  (Vr_  Grade.Triang]e)  x  t  _  Solidityi 

to  road-bed,  and  omitting  G.  T.  to  intersection  of  slopes. 

Where, 
B  =  Top-width  of  back,  or  larger  parallel  side  of  trapezoid, 

measured  horizontally. 

b  =  Bottom-  width  of  back,  or  lesser  side  of  trapezoid,  equal 

also  to  the  edge,  which  is  the  horizontal  top-width 
of  smaller  end  section,  at  a  distance  forward  =  to 
the  common  length  of  wedge  and  prism. 
H  and  h  =  Vertical  hights  of  the  end  sections  to  intersection  of 

slopes. 

H  —  h    =  Hight  of  back  of  wedge. 

r  =  Ratio  of  side-slopes  to  unity,  or  cot.  of  slope  angle. 

h2  r          =  Area  of  prism  to  intersection  of  slopes,  and  less  Grade 
Triangle  =  area  of  section  from  ground  to  road-bed. 
118 


CHAP.  V— FOURTH  METH.  COMP.— ART.  26.  119 

In  calculating  by  this  Formula  we  may  omit  the  Grade  Triangle 
if  we  choose  (though  we  should  have  to  supply  a  more  complicated 
expression  for  A2  r),  and  might,  perhaps,  somewhat  simplify  the  com- 
putation thereby;  but  if  used  in  area,  we  must  be  careful  to  account 
tor  it  in  volume;  while  the  bights  need  only  be  extended  from  ground 
to  road-bed  ;  though  as  their  difference  only  is  used  here,  that  is  not 
material — and  altogether  we  would  gain  so  little  by  the  change  as  to 
make  it  unadvisable. 

In  words,  this  Formula  ^ 

may  be  expressed  as  jot-  V  (Mean  Area  Wtdge  -f  Mean  Area  of 
lows :  )      Prism)  X  Common  Length  =  Solidity, 

of  the  Prismoid,  to  intersection  of  slopes, 
and  minus  G.  T.  to  Road-bed. 

Inasmuch,  however,  as  a  trapezoid  is  always  reducible  to  an  equiva- 
lent rectangle,  we  may  consider  this  matter  of  the  superposed  wedge 
in  a  more  general  manner,  without  the  necessity  of  first  reducing  the 
trapezoidal,  or  triangular,  cross-section  to  a  level  on  the  top,  or 
slope  of  0°. 

Before  entering  upon  this  branch  of  the  subject  we  may,  however, 
state  that  the  reason  why,  in  a  wedge  with  a  trapezoidal  back,  we 
sum  up  all  the  three  parallel  sides  of  back  and  edge  X  by  hight  of 
back  -i-  by  6,  and  finally  multiply  by  length  for  volume — is  drawn 
from  the  common  rule  for  a  wedge — (Twice  width  of  back  -j-  edge 
X  by  hight  of  back  -r-  by  6,  and  X  by  length  =  Volume.}  But  in  a 
wedge  with  a  trapezoidal  back — the  £  sum  of  top  and  bottom  parallel 
sides  X  2  =  simply  the  sum  of  those  parallel  sides ;  and,  as  in  an 
earthwork  solid,  the  lesser  parallel  side  also  (generally)  equals  the 
edge,  that  being  the  top  line  of  the  smaller  end  section,  situated  at  a 
distance  of  the  length  forward.  Hence,  B  +  b  •+•  b  is  usually  equiva- 
lent to  — —  X  2  -f  (b  the  length  of  the  edge) — which  will  be  found 
2i 

in  substance  as  a  term  in  Button's  Rule  for  wedges  (4to  Mens.,  1770)  ; 
but  more  concisely  expressed  in  Chauvenet's  Theorem. 

References  to  Fig.  79.* 

a  d       =  End  view  of  the  back  of  a  rectangular  wedge. 
af      =  Equivalent   parallelogram,   of    which  a  g   is   the  base, 
and  a  D  the  altitude. 

*  For  Figs.  £7  and  78,  see  Chapter  VII. 


120 


MEASUREMENT   OF   EARTHWORKS. 


a  D  =  Horizontal  projection  (7O71),  or  width  of  a  b  (the  back). 
a  I  =  Horizontal  projection  (35'36),  or  width  of  a  h  (the  edge) 
a  e  g  k  =  The  initial  square  of  50  square  feet  area,  which  is  con- 

707 


tained  in  the  back  = 

A  B  f  Vertical  and  horizontal 
C  D  \      rectangular  axes. 


50 


=  14*14  times. 


Fi    79. 


aedb   *,  Back  of  Wedge  _  area  =701. 

agfb     „  EqqiV:FaTall:     _    do.  «707. 

aegk  c=:  Initial  Square  _    do.  «  50. 

aeg^ 

S' =  Equal  ^Xa 
b/d  J 

ac »..^a  JHor.'projrofTjaclc 

al *  do:        ed£e. 


as.»« 


:o 


The  triangles,  a  eg  and  6  df,  are  identical,  and  the  one  cut  off,  and 
the  other  added,  make  the  two  parallelograms,  a  d  and  a/,  precisely 
equivalent  =  707  area,  for  each. 


CHAP.  V.— FOURTH  METH.  COMP.—  ART.  26.  121 

a  b  =  Width  of  back  of  rectangular  wedge,  inclined  at  an  angle 

of  45°  =  100. 
a  h  =  Width  of  edge,  or  top  of  forward,  or  smaller,  section  =  50. 


Now  (as  above  mentioned),  a  trapezoid  being  always  reducible  to  an 
equivalent  rectangle,  we  may  consider  in  this  place  the  superposed 
wedge  (with  reference  to  Fig.  79),  without  the  necessity  of  first  equal- 
izing the  end  cross-sections,  by  level  lines  on  the  top,  as  will  be  more 
clearly  seen  further  on. 

However  much  the  back  or  edge  of  a  rectangular  wedge  may  be 
inclined  from  a  level  plane,  the  resulting  volume  is  still  the  same  by 
using  their  projections  upon  the  horizontal  one  of  two  rectangular 
axes  (as  C  D),  instead  of  the  actual  widths  of  back  or  edge,  whilst  the 
hight  of  the  back  becomes  the  base  of  an  equivalent  parallelogram, 
of  which  the  projection  is  the  altitude  ; — this  will  become  evident  by 
reference  to  Fig.  79. 

For  example,  let  us  now  compute  the  wedge  shown  in  the  figure: 
1st,  As  though  it  were  upon  a  level,  and  the  back  a  rectangle.  2d, 
As  an  oblique  parallelogram  on  the  back,  and  inclined  at  45°  from  a 
level  line. 


1.  Rectangular  back — supposed  to  be  level.  Length  of  wedge  = 
100.  Breadth  of  back  =  100.  Edge  =  50.  Hight  of  back  = 
7-071. 

Here  we  have : — Sum  of  the  3  parallel  sides  of  edge  and  back  -r-  3. 

100 )_.-,  f        7-071      =  Altitude. 

100  }  -  100  =  Length. 

50    =  Edge.  Right  Section  1     2)707-100 


83i  =  Average  multiplier  *    .    .    =         83£ 

Volume    =  29,463  =  C.  Feet 

Computed  after  Chauvenet's  Theorem  (Geom.,  VII.  22). 


122  MEASUREMENT  OF  EARTHWORKS. 

2.  Oblique-angled  Parallelogram  for  Back,  and  inclined  45°.  Length 
of  wedge .=  100.  Hight  of  back  =  10.  Horizontal  projection  of 
back  =  70*71.  Horizontal  projection  of  edge  =  35'36. 

-Sum  of  the  3  parallel  sides  or  edges 
~1T~ 

70'71  1        -R    v  f        10  =  Altitude. 

70-71  I  "  100         =  Length. 

35-36        =  Edge.    Eight  Section  1     2^1000 
3)1778 


58-927      =  Average  multiplier     .  =      58-927 

Volume  =    29,463 

It  is  evident,  from  a  consideration  of  the  above  case  of  a  rectan- 
gular wedge,  whether  level  or  inclined,  that  the  same  process  would 
apply  to  the  trapezoidal  wedge  (usual  in  earthworks),  either  by  its 
reduction  to  an  equivalent  rectangular  one,  or  (when  diagrammed 
together)  by  projecting  both  sides  of  the  back,  and  also  the  edge, 
upon  the  horizontal  axis,  and  ascertaining  the  respective  lengths  of 
these  three  projections,  to  be  used  in  the  computation  of  volume,  by 
Chauvenet's  Theorem,*  instead  of  their  actual  measured  lengths, — this  is 
in  fact  the  method  of  the  engineer,  who  usually  disregards  the  incli- 
nation of  the  ground,  and  takes  all  his  measures  horizontally  and 
vertically. 

The  hight  of  the  back  of  the  inclined  wedge  being  in  the  case 
above,  ascertained  by  dividing  the  known  area  of  the  back  of  the 
rectangular  wedge,  by  the  Arithmetical  Mean  of  the  horizontal  pro- 
jections of  its  top  and  bottom  breadths ; — both  equal  in  the  above 
rectangular  back,  but  always  unequal  in  a  trapezoidal  one. 

With  these  preliminary  observations,  we  will  now  give  the  rule 
for  finding  the  volume  of  the  superposed  wqdge  in  ordinary  earth- 
works, with  examples  to  show  how,  by  the  simple  addition  of  the 
under-prism,  the  solidity  of  the  entire  earthwork,  between  any  two 
cross-sections  of  given  area,  and  distance  apart,  is  easily  ascertained,  in 
all  cases,  within  a  limit  hereafter  discussed  (Art.  29). 

27 Eulesfor  Computation  by  Wedge  and  Prism.     The  data 

required  to  be  given  will  be  as  follows  : 

*  Chauvenet's  Geoin.,  VII.  22  (Philada.,  1871). 


CHAP.  V.— FOURTH  METH.  COMP.— ART.  27.  123 

1.  Areas  of  end  cross-sections. 

2.  Distance  apart,  or  common  length  of  wedge  and  prism. 

3.  Sum  of  distances  out,  to  ground  edges  of  side-slopes, — which 
are,  in  fact,  the  projections  or  horizontal  widths  of  back  and 
edge,  as  well  as  the  right  and  left  distances  of  the  field  engineer. 

The  first  is  obtained  by  well-known  processes,  and  the  two  latter 
are  always  supplied  by  the  Field  Book  of  the  engineer. 

Then,  as  preliminary  steps:  (1)  Find  the  difference  of  the  areas 
of  the  end  cross-sections,  which  difference  is  the  area  of  the  back  of  the 
superposed  wedge.  (2)  Divide  this  difference  of  area  by  half  the 
sum  of  the  widths  of  the  back  (or  horizontal  projections),  which  gives 
the  vertical  mean  hight  of  the  back.  Now,  the  lower  side  of  the 
back  (when  both  sections  are  diagrammed  together)  equals  the  edge 
(or  top-width  of  the  smaller  end  section)  supposed  to  be  forward,  at 
a  distance  equal  to  the  common  length.  So  that  if  B  =  top-width  of 
larger  end  section,  —  b  will  equal  its  bottom  width  (and  also  that  of 
the  edge} — so  that  B  +  b  -p-  b,  for  the  wedge-shaped  part,  would  give 
the  sum  of  the  three  parallel  edges  (or,  in  reality,  their  horizontal 
projections)  to  be  divided  by  3,  for  use  in  ChauveneVs  Theorem. 

RULE. — When  the  width  of  the  large  end  is  equal  to  or  greater  than 
that  of  the  small  one. 

1.  Vertical  mean  hight  X  distance  apart  sections 

~2~ 

Sum  of  the  three  parallel  edges       T.  . 

5— a-  =  Volume  of  Superposed  Wedge. 

o 

2.  Smaller  end  area  X  length  (or  distance  apart  sections)  =  Vol- 
ume of  Prism. 

These  two  results,  added  together  =  Solidity  of  the  whole  Prismoid. 

a Prior  to  giving  examples  in  illustration  of  our  rule,  it 

appears  necessary  in  this  place  to  make  some  explanations  to  show 
the  generality  of  the  application  of  the  rule  drawn  from  Chauvenet's 
Theorem  (Geom.,  VII.  22)  for  the  volume  of  wedges. 

Wedges  are  always  formed  by  the  truncation  of  triangular  prisms, 
which  may  be  termed  their  elementary  body ;  and  are  usually  desig- 
nated by  the  outlines  of  their  backs — as  Rectangular,  Triangular, 
Trapezoidal,  etc. — The  Initial  Wedge  may  be  assumed  to  have  a  square 
back;  by  successive  transformations  of  which,  several  varieties  are 
easily  formed. 


124 


MEASUREMENT   OF  EARTHWORKS. 


(1)  Let  the  back  of  u  rectan- 
gular wedge  (or  the  initial  wedge) 
be  a  square,  on  a  side  of  6,  edge 
12,  length  20.— Then,  the  right 
section  =  (6  X  20)  -T-  2  =  60.— 
One-third  of  the  sum  of  the  lat- 
eral edges  =  (6  +  6  +  1 2)  -H 
3  =  8 ;  and  60  X  8  =  480  == 
Volume  of  the  Square  Wedge. 

(2)  Now,  suppose  the  edge  of 
(1)  to  be  contracted  to  a  point; 
then,  the  wedge  becomes  a  pyra- 
mid, for  which  case  the  rule  also 
holds; — thus,    right    section    = 


3  =  4;  and  60 


60 i  sum  of  edges  =  (6  -f  6  +  0) 

X  4  =  240  =  Volume. 

Proof:  By  the  common  rule  for  pyramids,  we  have,  base  (6 
X  6)  -*•  3  =  12 ;  and  X  by  altitude  20  =  240  =  Volume,  the 
same  as  before. 

(3)  Suppose  the  back  of  the 
square   wedge   (1)  to    be   con- 
verted into  an  isosceles  triangle, 
on  a  base  of  6,  and  hight  of  6 — 
other   dimensions   as   in    (1) — 

then  right  section  =  60 

i  sum  of  edges  =  (6  -f  0  -f  12) 
-r-  3  =  6 ;  and  60  X  6  =  360 
=  Volume. 

Proof:  Now,  the  inscription  of  the  isosceles  triangle,  within  the 
square  back,  evidently  cuts  off  two  pyramids,  of  which  the  volume 
of  each  =  (3X6)-f-2  =  9-v-3X20  length  X  2  in  number 
=  120  Volume,  of  pyramids  cut  away  from  the  square  wedge  (1)  ; 
— then,  480  —  120  =  360  =  Volume,  the  same  as  before. 

(4)  Now,  suppose  (1)  and  (2) 
to  be  placed  in  contact  sidewise, 
then  they  form  together  a  rect- 
angular wedge,  back,  12  by  6; 
edge,  12  ;  length,  20 :— right  sec- 
tion =  60 i  sum  of  edges 

=  (12  +12  +  12)  -3  =  12; 
and  60  X  12  =  720  =  Volume. 


CHAP.  V.— FOURTH  METH.  COMP.— ART.  27. 


125 


Proof:  By  two  Pyramids  =  (72  +-  3  X  20  =  480)  -f  (60  -*• 
3  X  12  =  240)  =  720,  the  same  Volume;  or,  by  addition  of  (1) 
and  (2)  =  480  -f  240  =  720,  Volume  as  before. 


(5)  Suppose  now  the  vertical 
sides  of  the  square  back  of  (1)  to 
close  in  gradually  until  they 
meet  and  coincide  in  a  single 
vertical  line ;  then  the  back  has 
vanished,  and  become  a  vertical 
edge,  while  the  original  one 
remains  horizontal,  dimensioned 
along  with  the  other  parts  as  in  (1) — and  we  have  right-section 

60 J  sum  of  edges  =  (12  -f  0  +  0)  -*-  3  =  4 ;  and  60 

X  4  =  240  =  Volume  of  this  peculiar  double-edged  wedge; 
which  is  composed  of,  or  may  be  decomposed  into,  two  pyramids, 
based  on  the  right-section,  as  common  to  both,  and  each  having 
an  altitude  of  half  the  edge,  or  6  (though  such  equal  division  of 
edge  is  not  essential) ;  hence,  we  may  assume  the  edge  12  to  be 

a  double  altitude;  and  (~  X  12)  =  240  —  Volume  of  both— 
the  same  as  before. 


(6)  Now,  suppose  the  vertical 
sides  of  the  square  (1)  to  become 
inclined  (at  any  angle  that  will 
not  extinguish  the  base  of  the 
back),  say  at  an  angle  of  i  to  1 
side-slope,  thus  reducing  the  base 
from  6  to  2,  then  we  have  the  right- 
section  as  before  =  60  ......  I 

sum  of  edges  =  (6  +  2  -f  12) 


6t  ;  and  60  X  6§  = 


400  =  Volume  of  Trapezoidal  Wedge. 

Proof:  In  this  case  two  triangular  pyramids  are  cut  away  from 
the  original  solid,  by  the  sloping  sides,  having  together  a  base  of 
4,  and  altitude  of  6  ;  then,  (6  X  4)  -4-  2  =  12,  which  -f-  3  and 
X  20  common  length  =  80  Volume  cut  away — but  Volume  of 
(1)  =  480  —  80  =  residual  Volume  =  400,  as  before. 

(7)  Now,  suppose  two  sides  of  the  square  back  of  (1)  to  gradu- 
ally reduce  their  contained  angle,  and  finally  to  vanish  upon  the 


126 


MEASUREMENT  OF  EARTHWORKS. 


diagonal — then  the  back  be- 
comes a  right-angled  triangle 
(the  side  joining  the  right-angle, 
say  perpendicular  to  the  edge), 
and  this  wedge  has  two  edges  (one 
original,  and  the  other  now 
formed  at  the  side  connecting 
with  the  acute  angle,  both  being  horizontal  edges).  Then,  the 

right-section  =  60 i  sum  of  edges  (6  -f  0  -f-  12)  -t-  3 

=  6 ;  and  60  X  6  =  360  =  Volume. 

Proof:  Divided  by  a  plane  diagonally  through  the  vertex  of 
the  triangular  back,  and  opposite  corner  of  the  edge,  we  may 
decompose  this  wedge  into  two  pyramids — the  one  with  a  base 
=  the  right-section  =  60,  and  altitude  =  the  original  edge  = 
12 ;  then,  60  X  12  -4-  3  =  Volume =  240 

The  other,  with  a  base  equal  to  the  triangular  back,  or 
(6  X  6)  -i-  2  =  18,  and  an  altitude  =  the  length  =  20 ; 
then,  18  -T-  3  =  6,  and  X  length  20  =  Volume  .  .  .  =  120 

Total  Volume  of  both  Pyramids =360 

the  same  as  before. 


(8)  A  Rhomboid  Wedge  is 
computed  in  a  similar  manner : 
— thus,  let  the  rhomboidal  back 
have  a  vertical  diagonal  =  12; 
the  other  =  4 ;  an  edge  of  12; 
length  =  20  ;  and  the  side-slopes 
being  $  to  1. 

Then,  the  right-section  = 
12  X  20 

~2~~ 
120  X  fr 


=120  ......  i  sum  of  edges, 

=  640  =  Volume. 


44-12 


;  and 


Now,  by  cutting  off  from  the  rhomboid,  near  the  lower  angle, 
any  given  triangle,  we  have  remaining  a  Pentagonal  Wedge. 

Thus,  suppose  we  cut  off  a  triangular  wedge  having  the  base 
of  its  back  uppermost  =  2  ;  altitude  =  3  ;  common  length  and 
edge  =  20  and  12. 

Then  its  right-section  =  ?-><  2°  -  2  +  12 


X 


=  140   Vol- 


ume, cut  off.    And  640  —  140 
Pentagonal  Wedge. 


500  =  the  Volume  of  the  residual 


CHAP.    V.— FOURTH    METII.   COMP.— ART.  28. 


127 


(9)  Let  us  now  consider  a 
Trapezoidal  Wedge— dimensioned 
like  (8),  with  side-slopes  of  i  to 
1,  forming  the  top  of  the  back, 
while  its  base  =  2. 

Let  one  side-high t  =  12  above 
intersection  of  slopes;  the  other 
=  6 ;  the  edge  =  12 ;  and  the 
length  =  20. 


Now,  we  may  compute  this  wedge  in  two  parts  as  follows: 

1.  As  a  triangular  wedge,  above  the  level  of  the  lowest 
side-hight. 


x 


12 


320 


2.  As  a  trapezoidal  wedge,  between  the  level 
mentioned  and  the  base  of  the  back. 


2+12 


Total  Volume 


180 
500 


Or,  as  in  (8),  we  may  compute  the  body  as  a  Ehomboidal 
Wedge,  and  deduct  the  triangular  wedge  cut  away  below  the 
base  of  2, — as  in  fact  we  did  in  (8), — the  resulting  volume  being 
500,  the  same  as  herein  found. 

Finally,  we  perceive  that  from  (1)  the  square  or  initial  wedge  we  may 
easily  deduce  several  varieties  of  wedges,  and  might  go  further. 

After  this  necessary  digression,  indicative  of  the  simplicity,  gen- 
erality, and  value  of  Chauvenet's  Theorem,  we  will  now  proceed  to 
illustrate  our  own  rule  (deduced  from  this  theorem),  as  applied  to 
Earthworks,  by  several  examples. 


28. 


Here  follows  the  calculation  of  some  examples. 


Example  1. — Computation  by  Wedge  and  Prism,  tested  by  Hights 
and  Widths,  under  Simpson's  Rule 


128 


MEASUREMENT  OF  EARTHWORKS. 
References  to  Fig.  80. 


In  this  case  equal  slopes  of  1  in  4  form  a  ridge  in  the  larger  end 
section,  and  a  hollow  in  the  lesser  one. 

Dimensioned  as  shown  in  the  figure  annexed. 


2000  area. 
JacliI  «12OO  area. 
Tadbl = mid.8ec.16OO  ar; 


-sr 


I.=ini:  of  sip: 


Data. 

Sq.  Ft. 

(  Differences  of  areas  of  end  sections  .........  =  800 

<  Widths,  or  horizontal  projections,  equal  for  both  sections     .  =    80 
I  Distance  apart  sections  ............     .  =  100 


To  find  the  vertical  mean  hight  of  back  of  wedge. 

C  2000  ) 
End  Areas  =  j  -^oo  f  Difference  of  Areas. 

Half  sum  of  widths  -—  80)  800 


10  =  Vertical  Mean  Hight  of  Back. 


CHAP.  V.— FOURTH  METH.  COMP.— ART.   28.  129 

Then,  by  the  Rule  above,  and  Chauvenefs  Theorem. 
Sum  of  3  parallel  sides  of  edge  and  back  -r-  3. 

80 )       -p     ,  f  f     Vertical  Mean 

}.  =  Back.  10       =  <  TJ.  ,  ,     /,  -r,     i 

80  j  (  Right  of  Back. 

80     =  Edge.   Right  Section  i  100  =     Common  length. 


3)240  2)1000 

80     =  Average  breadth  500  =     Area  of  right  sec. 

Right  section  X  Mean  breadth  =  500  X  80    .     .     .  =    40,000  =  Volume  of  Wedge* 
Smaller  end  area  =  1200  X  100,  length     .     .     .     .  =  120,000  =       "       "     Prism. 

Solidify  of  entire  prismoid =  160,000  Cubic  Feet. 

Proof,  by  Hights  and  Widths  (SIMPSON). 

Hights.     Widths. 

Larger  cross-section    .  =  50  X    80  =    4000  =26. 
Smaller     "         "        .  =  30  X    80  =    2400  =  2 1. 

Sums  of  his.  and  wids.  =  80  X  160  =  12800  =  8m. 
Divisor  —  12)19200 

"T600  =  Prism.  Mean  Area. 
100  =  Common  length. 
Solidity  of  entire  Prismoid  (as  above)  =  160,000  Cubic  Feet. 

Note. — By  BUTTON'S  General  Rule  we  have  the  same  Solidity  •-=* 
160,000  Cubic  Feet. 

Example  2. — Let  us  now  take  the  case  figured  for  another  purpose, 
by  Fig.  14,  Art.  8. 

Areas. 

Large  end  section -=  654  to  road-bed  only. 

Small     "         "       =  300    " 

Difference,  or   area  of  back  1  __ 
of  superposed  wedge     .     »    ) 

Supposing  the  smaller  end,  at  a  distance  of  100  feet  forward,  to  be 
ABKH  =  300  in  area.  While  the  larger  end  ABCDEFGHA  = 
654  area.  Common  length  =  100  feet. 

Widths. 
fU  _L    4ft 

Then,  -~--  =  47,  Mean  width  of  back. 

,  7-532  X  100  length  Ri^«on- 
and =  oTo'b 

=  7-532,  Vertical  Mean  Hight  of  Back. 

47 


130  MEASUREMENT  OF  EARTHWORKS. 

54  -4-  40  -f  40  =  Sum  of  the  three  parallel  sides 

—^r-  .  =  44f  feet 

o 

p.     „     (  376-6  X  44f    .     .     .  =  16822  =  Volume  of  Wedge. 
a  y>  \  300     X  100    length  ==  30000  =      "         "    Prism. 

Solidity  of  the  whole  Prismoid,  \       ~~ 

f  j  i  j  4  j  T  C  =  46822  =  Cubic  feet  to  road-bed. 

jrom  road-bed  to  ground  line        J 

or  56,822  to  inter- 
section of  slopes. 

Now,  roughly  computing  this  example,  both  by  Hights  and  Widths, 
and  by  Roots  and  Squares,  we  find  for  the  Solidity  about  the  same 
result,  the  difference  being  small  in  the  whole  body  of  earthwork  con- 
sidered. 

In  like  manner,  roughly  calculating  Figs.  43  and  44,  which  have 
very  irregular  ground  lines,  with  both  end  sections  in  each  case  dia- 
grammed upon  one  figure.  We  find  that  computed  by  Wedge  and 
Prism,  and  some  other  methods,  as  a  proximate  test,  they  all  coincide 
within  a  few  cubic  yards. 

So  that  this  rule  for  calculating  Prismoids  of  Earthwork  by  means 
of  a  Prism  and  Wedge,  superposed,  may  be  accepted  as  proximately 
correct  in  all  ordinary*  cases,  and  it  is  in  practice  a  very  simple  one, 
as  may  be  noticed  in  the  examples. 

Requiring  for  data  give,n  merely  the  areas  of  the  end  cross-sections, 
their  distance  apart,  and  their  total  widths  across,  horizontally,  to 
ground  edges  of  slopes  : — no  matter  how  irregular  the  surface  may  be. 

In  all  the  computations  above  (as  well  as  in  the  methods  of  pre- 
ceding chapters),  so  soon  as  the  mean  area  of  an  earthwork  solid  is 
ascertained,  it  will  be  found  conducive,  both  to  expedition  and  to 
accuracy,  to  resort  with  it  to  the  table  of  cubic  yards  for  mean  areas 
(at  the  end  of  the  book),  to  obtain  cubic  yards,  if  they  should  be 
required  in  the  resulting  volume. 

In  this  connection  it  may  be  observed  that  the  transverse  area  of 
the  under-prism  being  always  given  in  the  data  (and  usually  given  as 
that  of  the  smaller  cross-section),  whilst  the  distance  apart  sections 
is  also  known,  it  is  better,  where  cubic  yards  are  desired  in  the  ulti- 
mate solidity,  always  to  find  them  from  the  table  in  the  manner  shown 
by  the  directions  for  its  use;  and  the  superposed  wedge  may  be  also 
treated  in  a  similar  way  by  computing  its  mean  area. 

*  Where  the  cross-sections  appear  to  be  unusually  distorted,  so  as  to  render  doubtful, 
the  application  of  any  ordinary  rules,  then  we  must  endeavor  to  sketch  an  accurate  mid- 
section,  and  use  our  First  Method  of  Computation  (Chapter  II.) — which  never  fails  when 
the  data  is  correct. 


CHAP.  V— FOURTH  METH.  COMP.— ART.  29. 


131 


29 Although  the  foregoing  rule  for  the  computation  of  a 

Prismoid,  by  "Wedge  and  Prism,  is  proximately  correct  in  all  ordinary 
cases,  it  has  limits  which  must  be  observed,  when  exact  results  are 
sought. — These  limits  are:  That  the  extreme  horizontal  width  of  the 
smaller  end  section  shall  always  be  equal  to,  or  less  than,  that  of  the  larger 
end,  and  never  greater,  where  our  rule  is  used  as  written  above. 

Thus,  in  all  the  cases  computed  in  the  above  examples,  the  width 
of  smaller  end  is  less,  except  in  the  figure  next  preceding,  where  it  is 
equal — but  in  none  of  the  examples  is  it  greater,  and  hence  they  are 
all  clearly  within  the  limits  of  the  rule. 

In  the  following  figure  (Fig.  81),  however,  the  horizontal  width 
of  the  smaller  end  is,  in  this  unusual  case,  greater  than  that  of  the 


Pig.  81. 


larger  one — to  such  cases  then  our  rule  above  stated  does  not  apply 
directly  in  the  form  as  written. 


132  MEASUREMENT   OF  EARTHWORKS. 

A  consideration  of  the  figure  annexed,  where  both  end  sections  and 
the  mid-section  are  diagrammed  together,  will  make  the  reason 
evident. 

It  is  simply  this,  that  whenever  the  horizontal  top  line  of  the 
smaller  end  exceeds  in  width  that  of  the  larger  one,  or  lays  above  it 
(in  a  cut),  when  diagrammed  together  in  one  figure,  with  the  diedral 
angle  common  to  both,  then  the  smaller  end  ceases  to  be  the  section  of  a 
prism,  and  becomes  that  of  a  prismoid. 

But  as  a  prismoid  is  formed  of  an  under  prism,  with  a  wedge  super- 
posed, we  have  then  in  this  solid  (such  as  is  sectioned  in  Fig.  81)  a 
prism  with  two  wedges  superposed — the  upper  one  carrying  the  ground 
surface  of  the  earthwork  solid. 

The  prism  in  this  case  has  for  its  cross-section  the  portion  of  the 
solid  below  the  line  c  6,  marking  the  extreme  breadth  of  the  larger 
end  section,  while  the  two  superposed  wedges  are  reversed  in  position 
— that  in  contact  with  the  under  prism  having  its  edge  in  the  line  c  b, 
the  width  of  the  larger,  while  that  carrying  the  ground  surface  has  its 
edge  in  e  d,  the  width  of  the  smaller  end  section ;  and  therefore  the 
wedges  are  reversed  in  position,  though  having  the  same  length  in  com- 
mon with  the  prism,  which  underlies  both. 

Example  3,  Fig.  81. 

Cross-section  of  prism  below  c  b  =    400. 
"  "         smaller  end         =    900. 

Data  •{          "  "         larger  end  =  1200. 

Common  length  of  all  =  100  feet;  other  dimensions  as  in 
Fig.  81. 

(1)  By  Prwnoidal  Formula — First  Method  Computation,  Chapter 
II.  (Button's  General  Rule) — which  is  an  accepted  standard  for 
accuracy. 

Smaller  end  section  .     .     .  =    900      =  t. 
Larger     "         "         .     .     .  =  1200      =  b. 
Mid-section  deduced,  being 
a  mansard  figure  flat  on 
the  top  =  1425  X  4  .    .  =  5700      =  4  m. 

6)7800 
T300      ==  Prism.  Mean  Area. 

100  =  Common  length. 
Solidity    .......  =  130,000       Cubic  Feet. 


CHAP.  V.— FOURTH  METH.  COMP.— ART.  30.  133 

(2)  By  Chauvenet's  Theorem,  and  our  rule  drawn  from  it. 

I  (1)  =  The  top  wedge  (at  ground)  =  Right 
section  (40  X  100  -H  2  =  2000) 
X  i  sum  of  edges  =  (60  +  40 

+  0  -H  3  =  33i) =    66,667  C.  Feet- 

(2)  =  The  intermediate  wedge,  adjoining  the 
prism  (as  in  our  rule).  Difference 
of  areas  -f-  £  siim  of  widths  =  500 


I 

I 

S,  Then,  by  the  rule  (from  Chauve- 

net),  (10  X  100  -s-  2  =  500)  X 


50  =  10,  Mean  Hight  of  wedge. 
Tl 

a 


i  sum  of  edges  =  (60  +  40  +  40 

-f-  3  =  463) =    23,333    «    « 

(3)  =  The  prism,  which  underlies  both  = 

400  area  X  100  length  .     .     .     .  =    40,000    "     " 
Totality  of  this  solid,  containing  two 
wedges  and  one  prism  =  Solidity  =  130,000  C.  Feet. 


In  examining  the  solid  body  terminated  by  the  cross-sections  figured 
(in  Fig.  81),  it  will  be  found  to  be  bounded  upon  every  side  by  planes, 
passed  through  three  common  points,  so  connected  that  the  faces  con- 
tain no  warped  surfaces  whatever. 

30.  It  would  appear  that  in  peculiar  solids,  like  that  in  Fig.  81. 
we  might  omit  the  prism  entirely,  and  decompose  the  body  into  a 
species  of  double  triangular  or  rhomboidal  wedge  (with  base  of  back, 
and  also  the  edge,  common  to  two  triangular  wedges  superposed,  and 
inverted  with  their  bases  in  contact,  one  on  the  other),  and  this 
double  triangular  wedge,  with  a  single  pyramid  based  upon  the 
smaller  end  (or  in  fact  on  either  end),  all  having  a  common  length, 
would  form  the  whole  earthwork  solid,  and  simplify  the  calculation 
in  such  special  cases — if  not  in  all  cases  of  irregular  ground. 

Thus,  examining  the  large  end  I  b  a  c,  we  find  it  to  consist  of  the  backs 
of  two  triangular  wedges,  joined  together  at  their  bases  c  b,  and  hav- 
ing a  common  edge  at  100  feet  forward,  equal  to  d  e,  the  top  of  the 
smaller  end. 

Below  this  double  wedge  we  find  a  pyramid  whose  base  is  I  e  d  I,  and 
vertex  at  I,  with  the  common  length  of  100 — the  calculation  of 
solidity  is  as  follows: 


134  MEASUREMENT  OF  EARTHWORKS. 

Example  4  (Fig.  81). 

(1)   The  Double  (Triangular  or  Rhomboidal)  Wedge. 
The  mean  breadth  being  common  both  to  the  upper  and  lower  tri- 
angular part  of  the  larger  cross-section,  then  we  have,  — 

o 

-=  33*. 

And  the  whole  hight  of  the  double  triangular  wedge  is  composed 
of  the  hights  of  the  two  separate  parts  =  40  -f~  20  =  60,  forming  a 
Rhomboid. 

Then,  -  -^  -  =  3000  =  Right  Section. 

C.  Feet. 

And  right  section  =  3000  X  *  sum  edges  =  33*   .     .     .  =  100,000 
(2)   The  Pyramid,  based  on  smaller  end  =  —   X  100  .  =     30,000 

Solidity  of  the  whole  Prismoid =  130,000 

(Being  the  same  as  in  Example  3.) 

"We  might  also  divide  this  solid  into  two  wedges  and  a  pyramid  by 
other  cutting  planes,  with  the  same  result.  Thus : 

Example  5  (Fig.  81). 

Rt.  Sec.  %  sum  edges.  C.  Feet. 

(1)  lfe*r  l*dfc*L>^°?-  2000  x(4°  +  f  +  °)  =    66,667 

Rt.  Sec.  %  sum  edges. 

(2)  termed.  W^e,  5°*1°?  =  1500  x(60  +  43°  +  °)  =    50,000 

(3)  Pyramid  underlying  both  =  ---==  133*  X  100  length  =     13,333 

Solidity  of  the  whole  Prismoid =  130,000- 

(Being  the  same  as  in  Examples  3  and  4.) 

Suppose  now  upon  the  smaller  end  section  (Fig.  81)  we  place  a 
triangle  of  60  feet  base,  and  10  feet  altitude,  the  vertex  representing  the 
termination  of  the  crest  of  the  ridge  coming  from  the  apex  of  the  taller 
section,  and  thus  augment  the  area  of  the  lesser  end  to  an  equality  with 
the  other,  or  make  each  =  1200  in  area — the  addition  in  Solidity 
being  a  Pyramid. 

Then,  although  the  end  areas  are  now  equal,  the  horizontal  widths 
between  the  ground  edges  of  the  side-slopes  remain  unequal,  as  before ; 
the  big  end  having  least  width. 


CHAP.  V.— FOURTH  METH.  COMR— ART.  30. 

And  the  computation  of  this  solid  is  as  follows: 
Example  6  (Fig.  81). 


135 


x  4  =6000  =  4  m. 
6)8400 
1400  Pris.  Mean. 

100  Length. 
Sol.=  140,000  C.  Feet. 


By  known  Geometrical  Solids,  gov- 
erned by  Familiar  Rules. 

Pyramid  (super-added)  base  300. 
Then, 

300         Length. 

—  X  100 =    10,000 

(1)  Top  Wedge =    66,667 

(2)  Intermediate  Wedge   .     .     .  =    23,333 

(3)  Prism .     .  =    40,000 

Solidity  in  C.  Feet     .     .  =  140,000 


By  Hutton's  General  Rut 

,          =  1200  =  t. 
Find  Areas  .     . 

in,  The  mid-sec- 
tion deduced, 
being  a  man- 
sard figure, 
peaked  upon 
the  top  = 
1500  in  area. 

50  +  30  _ 

40  X  20  =  800 

?!2L_5=  75 

A  of  25'  =625 

~1500 


In  all  the  above  examples  (except  Example  2),  the  computation 
for  solidity  extends  from  ground  surface  to  intersection  of  slopes, 
without  regard  to  the  road-bed.  But  any  width  of  road-bed  may 
be  assumed,  the  volume  of  the  grade  prism  ascertained,  and  being 
deducted,  will  leave  the  solidity  from  road-bed  to  ground  all  the  same, 
as  if  it  had  been  specially  calculated  in  that  way. 

a Of  the  Rhomboidal  Wedge  and  Pyramid. 

A  close  examination  of  the  solid,  cross-sectioned  in  Fig.  81,  and 
shown  in  isometrical  projection  by  Fig.  82,  will  make  it  evident  that 
beginning  with  the  larger  end  section,  the  three  cross-sections  required 
by  HUTTON'S  General  Prismoidal  Rule  will  be  a  Rhomboid,  a  Penta- 
gon, and  a  Triangle,  dimensioned  as  shown  in  the  figures. 

And  the  solidity  of  this  body  by  BUTTON'S  Rule,  as  shown  in 
Example  3,  Art.  29  =  130,000  Cubic  Feet. 

It  is  also  evident,  from  Example  4,  of  this  article,  that  this  compu- 
tation can  be  made  for  solidity  with  the  same  result  (130,000  Cubic 
Feet),  by  decomposing  the  body  into  a  Rhomboidal  Wedge  and  two 
Pyramids,  which  may  be  aggregated  and  calculated  as  one,  so  that,  as 
in  Example  4,  this  solid  can  be  computed  as  though  it  were  composed 
of  a  single  Rhomboidal  Wedge,  having  its  edge  in  the  width  line  of 
the  smaller  end  section;  and  of  a  single  Pyramid  upon  a  base  equiva- 
lent to  the  latter  in  area,  and  its  vertex  at  the  foot  of  the  rhomboidal 


136 


MEASUREMENT  OF  EARTHWORKS. 


back  which  forms  the  area  of  the  larger  cross-section,  or  one  equiva- 
lent thereto,  and  standing  (as  both  end  sections  do)  with  the  vertices 
of  one  of  their  vertical  angles  coincident  with  the  line  of  intersection 
of  the  side-slopes  prolonged. 


Kg.  82 


Sea.  £0  TOT  Inch. 


By  means  of  Wedge  and  Prism,  or  Wedge  and  Pyramid  (especially 
the  latter),  we  have  already  indicated  the  process  of  reaching  the  vol- 
ume of  an  earthwork  solid,  and  we  will  now  continue  our  examples 
until  the  simple  combination  of  Wedge  and  Pyramid,  in  computing 
solidity  upon  the  usual  earthworks,  is  fully  illustrated. 

Although  solids  resembling  Fig.  81  in  their  cross-sections  admit  of 
being  easily  computed  by  their  own  dimensions,  either  by  Wedge, 
Prism,  and  Pyramid,  or  by  HUTTON'S  General  Rule,  which  is  a  stan- 
dard for  volume;  nevertheless,  as  earthwork  sections  generally  pre- 
sent themselves  in  a  somewhat  different  form,  it  becomes  desirable  to 
devise  a  rule  which,  within  a  long  range,  will  apply  to  all  earthwork 
with  uniform  slopes,  and  shall  include  within  its  limits  the  great 
majority  of  cases  which  come  under  the  notice  of  the  engineer. 


CHAP.  V  —FOURTH  METH.  COMP.— ART.  30.     137 

Extremely  irregular  and  distorted  solids,  however,  have  sometimes 
to  be  subjected^  to  calculation,  which  seem  almost  incommensurable 
by  any  fixed  rule,  and  such  exceptional  cases  must  be  left  to  inde- 
pendent methods  adopted  at  the  time ;  though  it  is  obvious  that  any 
solid  may  be  so  sectioned,  and  divided  into  limited  portions,  as  to 
admit  of  computation  by  many  processes,  without  material  error. 

b Statement.     In  any  earthwork  solid  contained  within  a 

diedral  angle  (formed  by  the  intersection  of  uniform  side-slopes), 
however  irregular  the  ground  may  be,  if  the  side-slopes  continue  uni- 
form— and  we  have  given,  the  length  I,  the  areas  of  the  cross-sections 
at  the  ends  A  and  A',  and  the  slope  ratio  r.  We  may  compute  the 
volume  of  such  solid  as  a  double  Triangular,  or  single  Rhomboidal 
Wedge  in  combination  with  a  single  Pyramid  (the  latter  also  usually 
Rhomboidal  but  sometimes  Triangular). 

Process. — Take  any  pair  of  irregular  cross-sections,  judiciously 
located  and  measured  by  the  field  engineer,  so  as  correctly  to  define 
the  ground,  and  of  which  all  the  necessary  dimensions  are  known,  as 
well  as  the  distance  apart  sections. 

1.  Ascertain  the  areas  of  the  cross-sections  to  intersection  of 
side-slopes. 

2.  Find  the  proper  hight  from  intersection  of  slopes,  to  include 
one-half  the  area,  also  the  proper  width,  and  assume  this  as  the 
base  of  the  back  of  a  double  Triangular,  or  Rhomboidal  Wedge 
in  the  larger  end,  and  as  the  edge  of  the  same  in  the  smaller  one. 

3.  Compute  from   the   larger,  or   from   either  end  section,  a 
Rhomboidal  Wedge,  by  Chauvenet's  Theorem.     (See  Example, 
Art.  27,  a,  paragraph  8.) 

4.  Then,  to  the  solidity  of  this  Rhomboidal  Wedge,  add  that 
of  a  Pyramid,  based  upon  the  other  end  section,  and  having  for 
its  altitude  the  common  length,  or  distance  apart  sections.     (See 
rule  following.)    . 

The  sum  of  the  altitudes  of  the  double  triangles  (joined  at  their 
bases)  forms  the  vertical  diagonals,  or  hights  of  back,  of  the  rhomboi- 
dal  wedges,  while  their  horizontal  "diagonals  form  the  width  of  back 
at  one  end,  and  of  the  edge  at  the  other,  the  angular  points  of  the 
Rhomboid,  vertically,  being  zero.  Either  end  may  be  calculated  from, 
while  the  other  area  is  the  base  of  a  pyramid  (Rhomboidal,  Triangu- 
lar, or  Irregular),  having  for  altitude  the  common  length  I.  For 
proof  of  the  work  we  should  always  make  both  direct  and  reverse  calcu- 


138  MEASUREMENT  OF  EARTHWORKS. 

lations,  taking  either  end  alternately  as  the  base,  and  though  they 
will  seldom  agree  exactly,  owing  to  the  decimals  coining  in  a  different 
order  (unless  we  use  a  cumbrous  number  of  places)  ;  nevertheless, 
the  agreement  will  be  found  close  enough  for  a  verification  of  such 
work. 

To  compute  the  Rhomboidal  Wedge  and  Pyramid  in  an  Earthwork. 
Adopt  either  end  for  Base,  and  call  the  other  the  Top  =  b  and  t,  of 
former  notations. 

Present  notation  : 

A.  —  Area  of  cross-section  assumed  for  the  Base. 
A!  =      "  "  "  "  "    Top. 

I     =  Common  length,  or  distance  apart  sections. 

These  are  all  the  data  required  to  be  given,  the  remainder  needed 

are  easily  computable. 

h    \  Vertical  diagonals  of  the  equivalent  Rhomboids,  into  which 
h'  J      the  end  areas  are  transformed. 


,  >  Horizontal  diagonals  of  the  same. 
Then,  by  computation  : 


From  the  foregoing  it  is  evident  that  w  =  h  r,  and  w'  =  h'  r. 
Also,  when  the  slopes  are  1  to  1,  then  h  =  \/2  A;  if  1£  to  1, 
h  =  V-fAT;  and  if  2  to  1,  h  =  VAT  The  use  of  these  will  often  be 
convenient. 

RULE.  —  Case  1.  —  Where  width  of  big  end  is  equal  to,  or  greater 
than,  that  of  small  end. 

1  (Half  product  of  vertical  diagonal  of  base,  by  distance  apart 
sections)  X  (One-third  the  sum  of  horizontal  diagonals  of 
both  ends)  =  Solidity  of  Rhomboidal  Wedge  ; 


CHAP.  V.—  FOURTH  METH.  COMP.—  ART.  30.  139 

2  (One-  third  of  area  of  top)   X   (Distance   apart   sections)  = 
Solidity  of  Pyramid  ; 


3.  Add  together  the  two  solidities  above  (1  and  2)  for  the  solidity 
of  the  entire  Prismoid  :  —  from  ground  to  intersection  of  slopes, 
and  minus  the  volume  of  the  grade  prism,  gives  solidity  from 
road-bed  to  ground. 

RULE.  —  Case  2.  —  Where  width  of  big  end  is  equal  to,  or  less  than, 
that  of  small  end. 

In  this  case  the  multiplier  for  edges  (No.  1,  Case  1)  is  to  be 

(w  +  w')  +  (w  —  w)  .  (w  +  «0      ,„,  .. 

-,  instead  of  simply  -  —  -  .  .    While  to 
6  6 

the  volume  produced  by  the  Rule  of  Case  1  —  modified  in  the 
multiplier  as  just  mentioned  —  we  must  add  a  final  correction, 
as  follows  :  (Difference  of  actual  horizontal  widths  X  Difference 
of  their  hights  from  intersection  of  slopes)  X  length  —  this 
final  product,  added  to  the  volume  resulting  from  the  rule  above, 
gives  the  solidity  for  Case  2. 

The  application  of  these  corrections  will  be  shown  hereafter  by 
an  example,  drawn  from  the  peculiar  solid,  figured  in  Figs.  81 
and  82. 

The  results  produced  by  these  corrections,  when  added  to  those 
obtained  by  the  Rule  of  Case  l,will  give  the  solidity,  whenever 
the  actual  width  of  the  smaller  end  section  does  not  exceed  three 
times  that  of  the  greater  one. 

Within  these  limits  the  rules  and  corrections  above  will  apply,  and 
they  will  be  found  to  cover  the  great  majority  of  practical  cases;  but 
where  thl  end  sections  are  even  more  distorted,  we  must  then  com- 
pute by  Mutton's  General  Rule,  or  by  the  actual  dimensions  of  the 
solid,  decomposing  it  into  elementary  bodies. 

As  the  Prism,  Wedge,  and  Pyramid,  are  the  solid  elements  from 
which  every  great-lined  body  is  composed,  and  into  which  it  may  be 
again  resolved,  it  follows  by  parity  of  reasoning  (as  in  the  case  of  the 
Prismoidal  Formula)  that  for  all  earthwork  solids,  bounded  by  planes, 
the  rules  of  this  chapter  hold. 

C  ......  We  will  now  illustrate  our  method  of  Wedge  and  Pyra- 

mid, by  computing  the  cases  of  Chapter  II.,  figured  from  53  to  64 
inclusive,  and  all  originally  computed  by  HUTTON'S  General  Rule- 
the  standard  for  accuracy. 


140  MEASUREMENT  OF  EARTHWORKS. 

All  of  these  examples  (as  indeed  is  the  fact  with  most  others  in 
practice)  come  under  our  Rule  and  Case  1  —  the  width  of  the  larger 
end  section  being  in  every  instance  greater  than  that  of  the  smaller 
one.  (See  Figs.  53  to  64,  Art.  18. 

Art.  18.  —  Example,  illustrated  by  Figs.  53  to  55. 

Given  areas  f  b  =  990  =  A  ^  Vertical  diago-  (  h  =  44-50  1  Horizontal  dia-  (  w  =  44'50  j 
to  intersection  It  •=  500  =  A'  [•  nals  computed.  \  h'  =  31-62  J  gonals  computed.  \wf  =  31-62  J 
of  slopes,  etc.  (  I  =100  feet.  J 

The  road-bed  being  20  feet  ;  the  side-slopes  1  to  1  in  this  case,  as 
in  all  where  r  =  1  ;  the  Rhomboid  becomes  a  square,  and  the  diago- 
nals equal. 

Direct  calculations. 

h  X   I  w  +  *?          o 

—  ~     X    —  —    =  S'  ° 


44-50  X  100  44-50  +  31-62 

—  -g  -    X  —  g  -      .  •  .  =  56,471 

A7 

—  -  x  I  =  &  of  Pyramid. 
o 

500 

-^   X  100    ...........  =  16,667  =  Pyramid. 

Total  .    ......  .    ..  >    .    .  =  73,138       C.  Feet. 

Deduct  Grade  Prism  ........  =  10,000 

Leaves  Solidity  of  Earthwork    .....  =  63,138 
As  computed  in  Art.  18,  Chapter  II.     .     .  =  63,170 

Difference    .........  =    —  32 

Reverse  calculations. 

31-62  X  100   -  31-62  +  44-50 

—  ^  -   X  -  g  -  .     .     .     .  =  40,126  ==  Wedge. 

QQA 

~    X  100  ...........  =  33,000  =  Pyramid. 

Total  .....  .   ......  =  73,126        C.  Feet. 

Deduct  Grade  Prism  ........  •=  10,000 

Leaves  Solidity  of  Earthwork    .....  =  63,126     ' 
As  computed  in  Art.  18,  Chapter  11...  =  63,170 

Difference  .........  =    —  44 

The  above  example  represents  an  earth-cut  upon  three-level  ground. 


CHAP.  V.— FOURTH  METH.  COMP.— ART.  30. 
Art.  18. — Example,  illustrated  by  Figs.  56  to  58. 


141 


This  example  represents  an  earth-cut  on  -five-level  ground,  having  a 
road-bed  of  20 ;  slopes  of  1  to  1 ;  length  100  feet. 

Computed  by  our  Rule,  Case  1,  we  have. 


Direct  calculations. 

Wedge     .    .  =  24,306 

Pyramid  .     .  =  14,367 

38,673 

Deduct  G.  P.  =  10,000 

Solidity  .  =  28,673 
By  Art.  18  .  =  28,650 

Difference.  =    +  23  C.  Feet. 


Reverse  calculations. 

(Wedge     .     .  =  27,254 
Pyramid.     .  =  11,467 
38,721 
,  Deduct  G.  P.  =  10,000 
j      Solidity.     .  =  28,721 
[  EyArt.  18  .  =  28,650 
\     Difference.  =    -f  71  C.  Feet. 


Art.  18. — Example,  illustrated  by  Figs.  59  to  61. 

This  example  represents  an  earth-cut  on  seven-level  ground,  dimen- 
sioned as  above.  . 

Computed  by  our  Rule,  Case  1,  we  have: 


Direct  calculations. 


Reverse  calculations. 


Wedge     .     .  =  42,048 
Pyramid  .     .  =*  21,700 

Wedge 
Pyramid 

Deduct  C 
Solidity 
By  Art.  ] 

Differe 

63,748 
Deduct  G.  P.  =  10,000 
Solidity.     .  =  53,748 
By  Art.  18  .  =  53,733 

Difference  =    +  15  C.  Feet. 

42,935 
20,800 
63,735 
10,000 
53,735 
53,733 

-f    2  C.  Feet. 


Art.  IS.— Example,  illustrated  by  Figs.  62  to  64. 

This  example  represents  an  embankment  upon  nine-level  ground, 
very  rough.     Road-bed  16  feet;  side-slopes  1£  to  1;  length  100  feet. 


Areas  given  f  t  = 
to  intersection  \  b  = 
of  slopes,  etc. 


iven  f  t  =  828%  =  A  ")  Vertical  diag 
tion  \  b  =  644%  =  A'  I  nals  computed, 
tc.  (I  =100  feet.  J 


Vertical  diago-  f  h  =  33-24  1        Horizontal  dia-  f  w  =  49-86  1 
h'  =»  29-<W  J    gonals  computed.  |  «/  =  43-98  » 


142  MEASUREMENT  OF  EARTHWORKS. 

Direct  calculations. 


33'24  ><  10°  X  £ =  51,987  Wedge. 

2  o 

644'67  X  100  .    ;'v  v    •  "•'  -*kiv  V  *     .     .  =  21,489  Pyramid. 

73^476 
Deduct  Grade  Prism.     .     .     .'..'.     .     .  =    4,267 

Solidity .     .     .  =  69,209  C.  Feet. 

As  computed  in  Art.  18,  Chapter  II.   .     .     .  =  69,200 

Difference  .  ...    C  "!'.  =      ~+~9  C.  Feet. 


Reverse  calculations. 

29-32  X  100  •     49-86  +  43-98 

2 3 =  45,856  Wedge. 

828-67 


3 


X  100 .....=  27,622  Pyramid. 


73,478 

Deduct  Grade  Prism.  -  ." =    4,267 

Solidity =  69,211  C.  Feet. 

As  computed  in  Art.  18,  Chapter  II.   ...  =  69,200 

Difference =    -f  11  C.  Feet, 

t 


d We  have  thus  compared  the  whole  four  of  the  examples 

illustrated  in  Chapter  II.,  and  all  computed  by  HUTTON'S  General 
Rule.  These  we  find  to  agree  with  the  calculations  by  Wedge  and 
Pyramid,  in  every  instance  within  a  few  cubic  feet,  and  had  the  deci- 
mals (into  which  all  these  computations  run)  been  carried  further, 
the  agreement  would  probably  have  been  closer. 

We  will  now  compute  by  Wedge  and  Pyramid  the  example  of  a 
heavy  embankment,  taken  from  Warner's  Earthwork,  Art.  86. 

"  Prismoid.     First  end-hight  —  28'7 ;  second  end-hight  — 14'5 ; 
surface-slope  15° ;  side-slope  H  to  1;  road-bed  24  feet." 


Data  computed  f  b  —  2411  =  A  ~\  Vertical  diago-  f  h  =  56-70  j  Horizontal  dia-  f  w  =  85-05  J 
to  intersection  of  -I  t  —  907  =  A'  >•  nalg  computed.  {  h'  —  34'78  j  gonals  computed.  (  w'  =  5217  } 
slooes.etc.  U=  100  feet.  ) 


CHAP.  V.— FOURTH  METH.  COMP.— ART.  30.  143 

Direct  calculations. 

56'7°  9X  10°     X    **+«*»    .     .    .  =  12%  Wedge. 

L  o 

-  X  100 =    30,233  Pyramid. 

159,906 

For  Cubic  Yards  -=-27 =      5,923 

Deduct  volume  of  Grade  Prism =          356 

Solidity =   ~5,567C.  Yards. 

By  Hutton's  General  Rule =      5,566 

\  Difference =         +  1  C.  Yard. 


Reverse  calculations. 


34-78  X  100  52-17  +  85-05  £*«* 

—    X  —      .    .    .  =    7y,o4J 


<  100    . =    80,367  Pyramid. 

Jf  9,909. 

For  Cubic  Yards  -r-  27 =  5,923 

Deduct  volume  of  Grade  Prism =  356 

Solidity =  5^67 

By  Hutton's  General  Rule =  5,566  . 

Difference =        -f  1  C.  Yard. 

Mr.  "Warner  (in  Art.  86  quoted)  makes  the  volume  here  computed 
:  5562  Cubic  Yards. 


6 All  of  the  above  examples  come  under  Case  1,  of  our 

Rule,  as  ordinary  earthwork  sections  usually  do.  But  we  will  now 
compute  a  single  example  by  Case  2 — where  the  width  of  the  greater 
end  is  less  than  that  of  the  smaller  one.  This  condition  will  be  found 
in  the  solid  figured  in  Figs.  81  and  82. 

In  this  example,  illustrative  of  the  rule  in  Case  2,  the  corrections 
therein  named  have  been  duly  embodied. 


144  MEASUREMENT  OF  EARTHWORKS. 

Example  of  Case  2  (Fig.  81).* 
4W8_XJOO  x  4M>8+  4*42  +  6-66  _    _  = 

h  X  I  (tfl  +  tip  +  (M  —  i*Q 


QAA 

irr  X  100  ............  =    30,000  Pyramid. 

o  - 

=  ^  x  i.  110,000 

Final  correction,  10  X  10  X  20  X  100  .     .  =    20,000 

Solidity  .    .    .    .     .     .    .    "...     .  =  130,000  C.  Feet. 

The  same  as  computed  before  ......  =  130,000 

It  would  appear,  then,  from  the  discussion  in  this  chapter,  the 
examples  given,  and  the  simplicity  and  conciseness  of  the  rules  for 
computing  earthworks,  by  means  of  the  Prism,  Wedge,  and  Pyramid, 
that  they  deserve  to  rank  amongst  the  best  employed  for  the  purpose. 

*  Although  this  solid  (Figs.  81  and  82)  is  bounded  on  all  sides  by  plane  surfaces,  and 
is  composed  simply  of  a  Rhomboidal  Wedge,  superposed  upon  a  Pyramid  —  very  few  of 
the  E,ules  or  Tables,  of  the  numerous  writers  on  Earthwork,  furnish  means  for  comput- 
ing its  solidity  —  which  can  only  be  readily  ascertained  by  BUTTON'S  General  Rule,  or  by 
decomposition  into  elementary  solids,  of  which  the  rules  for  volume  have  been  long 
established. 


. 

=r sp* 


CHAPTER    VI. 

PROFESSOR  GILLESPIE'S  FOUR  USUAL  RULES,  WITH  THEIR  CORREC- 
TIONS, AND  A  COMPARISON  OF  HIS  CHIEF  EXAMPLE  WITH  OUR 
THIRD  METHOD  OF  COMPUTATION — OR  ROOTS  AND  SQUARES  (CHAP- 
TER IV.). 

31 The  late  Professor  IV.  M.  Gillespie,  of  Union  College, 

Schenectady,  N.  Y.,  was  an  able  teacher  of  Civil  Engineering,  and  a 
sound  practical  writer  on  that  and  cognate  subjects,  as  may  witness 
his — Roads  and  Railroads  (1847),  10  editions;  Land  Surveying 
(1855),  8  editions;  Higher  Surveying,  etc.  (1870),  posthumous,  1 
edition ;  and  numerous  valuable  papers,  read  before  the  American 
Scientific  Association,  or  printed  in  scientific  journals. 

In  1847  he  published  his  first  edition  of  Roads  and  Railroads,  and, 
as  an  appendix  to  it,  in  about  25  pages,  he  gave  a  practical  summary 
of  various  methods  of  computing  Excavation  and  Embankment, 
accompanied  by  valuable  corrections  and  suggestions,  which  were 
together  so  explicit  and  so  well  grounded  that  this  Appendix  has 
become  the  basis  of  several  works  upon  the  subject,  whose  authors, 
without  much  acknowledgment  (often  without  any),  have  freely 
availed  themselves  of  Professor  Gillespie's  labors. 

His  work  on  Roads  and  Railroads,  well  printed  and  cheaply  pub- 
lished, has  had  a  great  circulation;  it  has  already  filled  10  editions, 
and  is  probably  better  known  in  the  offices  of  engineers,  all  over  this 
country,  than  any  other  similar  book.  In  the  Appendix,  on  Excava- 
tion and  Embankment,  Professor  Gillespie  recognizes  "four  usual 
methods  of  calculation" 

1.  Calculation  by  Averaging  End  Areas  (or  Arithmetical  Average). 

2.  "         "         Middle  Areas. 

3.  "        "         Prismoidal  Formula. 

4.  "  Mean  Proportionals  (or  Geometrical  Average). 

And  we  will  now  proceed  to  give  his  views  substantially,  but  not 
literally,  upon  these  four  rules,  which  he  found  in  use  when  he  took 
up  this  subject  in  1847,  and  which,  indeed,  had  long  before  been  known, 
— as  follows: 

10  145 


146  MEASUREMENT  OF  EARTHWORKS. 

1st.  Arithmetical  Average. — This  consists  simply  in  adding  together 
the  areas  of  any  two  adjacent  cross-sections,  taking  half  their  sum 
for  a  mean  area,  and  multiplying  it  by  the  length  of  the  station,  or 
distance  apart  sections, — to  find  the  Solidity. 

As  generally  used  by  engineers,  instead  of  adding  the  end  areas, 
halving  their  sum,  etc.,  they  employ  the  sum  of  the  two,  or  double 
areas,  and  merely  double  one  of  the  divisors  in  working  for  Cubic 
Yards,  as  follows : 

Engineers'  Rule. 

(Take  the  sum  of  the  areas  of  any  two  adjacent  cross-sections, 
multiply  these  double  areas  by  the  length  (which,  if  a  full  station 
\  of  100  feet,  is  done  mentally,  or  by  removing  the  decimal  point 
/  two  places  to  the  right).     Divide  by  6  and  by  9,  and  the  last  quo- 
\  tient  gives  the  volume  in  Cubic  Yards.          <A 

This  Rule  has  been  by  far  the  most  used  of  any  other  in  our  coun- 
try ; — with  tables  of  Cubic  Yards,  for  double  areas,  it  is  very  expedi- 
tious, and  has  found  numerous  advocates  amongst  engineers  on 
account  of  its  simplicity  and  convenience ;  it  usually  gives  a  result 
in  excess  of  the  truth,  and  where  the  disparity  of  areas  is  great,  very 
much  in  excess;  even  this  well-known  error  has  found  commendatory 
advocates,  on  the  ground  that  it  is  like  the  merchant  giving  good 
measure  to  the  customer,  and  that  this  excess  in  quantity  being  well 
understood,  would  be  compensated  for  by  a  reduced  price,  whenever 
the  work  was  executed  by  contract — but  these  arguments  are  clearly 
unsound. 

Professor  Gillespie  has,  however,  indicated  a  simple  correction,  by 
means  of  which  the  result  of  a  computation,  by  Arithmetical  Aver- 
age can  be  reduced  to  the  truth. 

Thus,  let 

d   =  Difference  of  centre  hights,  supposing  all  the  cross-sections  to 

be  reduced  to  an  equivalent  level  top. 
3*  •-=  Ratio  of  the  side-slopes  (or  cot.  of  angle)  s  to  1. 
I    =  Length  of  the  cut  or  fill  between  sections. 

*  Engineers  and  writers  have  pretty  generally,  of  late  years,  agreed  to  designate  the 
ratio  of  side-slopes  as  r  (and  this  we  have  usually  employed),  while  the  symbol  «  is  con- 
fined to  slopes  of  ground,  or  surface  slopes,  but  in  the  present  case  Professor  Gillespie's 
notation  is  adhered  to. 


CHAP.  VI.—  GILLESPIE'S  RULES,  ETC.— ART.  31.        147 


s  d?  I 
Then,  — ^—  is  the  proper  correction  for  the  results  of  Arithmetical 

Average,  which  correction,  if  computed  for  each  mass  so  calculated, 
and  then  deducted  therefrom,  will  give  the  true  solidity — the  same  pre- 
cisely as  if  calculated  direct  by  the  Prismoidal  Formula  itself. 

The  chief  example  computed  by  Professor  Gillespie  under  the  sev- 
eral heads  of  his  subject,  has  the  same  data  in  all,  as  shown  by  the 
first  four  columns  of  the  following  Tables — the  cross-sections  in  all 
cases  being  assumed  to  be  equivalent  level  trapezoids  by  him. 

1.  Arithmetical  Average. 

Table  1,  computed  in  illustration  of  the  corrections  proposed, 
including  an  entire  section  of  a  supposed  railroad,  4219  feet  in  length. 

1.'  Road-bed  50;  side-slopes  of  excavation  1J  to  1;  of  embank- 
ment 2  to  1. 


Sta. 

~r 

2 
3 
4 
5 

6 
7 

t  — 

Dis- 
tance 
in 
feet. 

Cut. 
+ 
in 
feet. 

¥ 

20 
0 

-sT 

Fill. 

in 
ft. 

0 

19 

8 

_o 

27 

End 
Areas, 
or 
QAM* 

sees. 

Excava- 
tion. 
C.  Feet. 

Em- 
bank- 
ment. 
C.  Feet. 

CORRECTIONS. 

Corrected  quanti- 
ties, agreeing 
with  the 
Prismoidal 
Formula. 

Sq.  Ft. 

Computed 
by 
Arith.  Average. 

By  Formula 
scP  I 
~6~ 

Amounts  in 
Cubic  Feet, 
deductive. 

Excava- 
tion. 
C.    Feet. 

Embkt. 
Cubic 
Feet. 

561 

858 
825 
820 
825 
330 

o 

1386 
1600 
0 
1672 
528 
0 

388,773 
1,280,994 
660,000 

685,520 
907,500 
87,120 

HXIS'XSGl 

45,441 

858 
82,500 

128,79t) 

98,673 
33,275 

7,040 
13pS8 

343,332 
1,280,136 
577,500 

586,847 
874,225 

80,080 
1,541,152 

9 

UX    2'X858 

6 

li  X  203  X  825 

6 

2    X199X820 

6 

2    X  Hs  X  825 

6 
2    X    8"X33° 

4219 

+  2986 
—  2200 

2,329,767 

1,680,140 

6 

2,200,9ti8 

From  this  Table  it  will  be  perceived  that  the  error  of  the  process 
of  Arithmetical  Average,  in  this  example,  amounts  in  Excavation  to 
6  per  cent.,  and  in  Embankment  to  9  per  cent.,  above  the  true  solidity. 

2d.  Calculation  by  the  Middle  Areas. — The  second  method  of  calcu- 
lation is  to  deduce  the  middle  areas  (commonly  called  mid-sections) 
of  each  Prismoidal  mass,  from  the  middle  hight,  or  Arithmetical 
Mean  of  the  extreme  hights  of  the  solid,  and  multiply  the  middle 
area  thus  found  by  length  for  volume.  The  results  thus  obtained  are 
too  small;  their  deficiency  being  equal  to  just  half  the  excess  of  the 
first  method. 


148 


MEASUREMENT  OF  EARTHWORKS. 


s  d2 1 

Here  the  corrective  formula  is,——;  and 

3.22 


corrections   thus   calcu- 


lated being  added  to  the  results  obtained,  by  the  process  of  middle 
areas,  would  make  them  coincide  with  the  true  volume  given  by  the 
Prismoidal  Formula. 

2.  Middle  Areas. 

Table  2,  computed  and  corrected  in  illustration  of  the  above,  including- 
an  entire  section  of  a  supposed  railroad  =  4219  feet  in  length. 

2.  Road-bed  50 ;  side-slopes  of  excavation  1 J  to  1  ;  of  embank- 
ment 2  to  1. 


Sta. 

Dis- 
tance 
in 

feet. 

Cut. 

i 

feet. 

Fill. 

in 
ft. 

Middle 
Areas. 

Computed 
by 
Middle  Areas. 

CORRECTIONS. 

Corrected   quanti- 
ties, agreeing 
with  the 
Prismoidal 
Formula; 

Exca- 
va- 
tion. 

Em- 
bank- 
ment. 

By  Formula 
sd*l 

"12" 

Amounts  in 
Cubic  Feet, 
additive. 

Ex- 
cava- 
tion. 

Em- 
bank- 
mont. 

Sq.  Ft. 

Cubic  Feet. 

Ex. 

Em. 

C.    Feet. 

C.    Feet, 

1 
2 

3 

4 
5 

6 

7 

561 
858 
825 
820 
825 
330 

8 

20 

0 

o 

19 
8 
0 

571-5 
1491-5 
650 
655-5 
1039-5 
232 

320,611 
1,279,707 
536,250 

537,510 
857,587 
76,560 

liX!SaX561 

22,721 
429 
41,250 

49,337 
16,638 
3,520 

343,332 
1,280,136 
577.500 

586,847 
874,225 
80,080 

12 

1JX    2s  X  858 

12 

liX293X825 

12 
2    X  19*  X  820 

12 

2    X  11s  X  825 

12 
2   X    89  X  330 

4219 

38 

27 

+2713-0 
—1927-0 

2,136,568 

1,471,657 

12 

64,400 

69,495 

2,200,968 

1,541,152 

From  the  above  Table  it  will  be  perceived  that  this  process  of 
Middle  Areas  is  a  closer  one  than  that  of  Arithmetical  Average ;  but 
being  in  deficiency,  while  the  former  was  in  excess,  the  difference  in 
this  case,  from  the  true  solidity,  being  about  3  per  cent,  less  in  Exca- 
vation, and  about  4  per  cent,  less  in  Embankment, 

3d.   Calculation  by  the  Prismoidal  Formula. — The  mass  of  which  the 
volume  is  demanded  is  a  true  Prismoid,  and  its  contents  will  there- 
fore be  given  by  the  well-known  Prismoidal  Formula. 
b  +  4m  -f  t 


6 


Where, 


X  length  =  Volume. 

b   =  Area  of  Base. 
m  =  Mid-section. 
t    =  Area  of  top. 


CHAP.  VI.— GILLESPIE'S  RULES,  ETC.— ART.  31.        149 

Retaining  the  same  data  for  the  example  as  has  been  used  in  the 
preceding  tabulations,  and  will  be  continued  throughout  this  discus- 
sion, we  refer  to  the  following  Table  (3),  where  the  results  obtained 
from  the  data  given,  by  means  of  the  Prism oidal  Formula,  are  pro- 
perly tabulated. 

3.  Prismoidal  Formula. 

Table  3,  in  illustration  of  the  computation  by  it.  Including  an 
entire  section  of  a  supposed  railroad  =  4219  feet  in  length. 

3.  Road-bed  50 ;  side-slopes  of  excavation  1  £  to  1 ;  of  embank- 
ment 2  to  1. 


Dis- 

Mid- 

QUANTITIES. 

Sta. 

tance 

in 

Cut. 

+ 

Fill. 

End 
Areas. 

dle 
Areas. 

Excava- 

Embank- 

tion. 

ment. 

feet. 

Sq.Ft. 

Sq.  Ft. 

C.  Feet. 

C.  Feet. 

1 

2 

561 

i 

+1386 

+  571-5 

343,332 

3 

858 

20 

+1600 

+  1491-5 

1,280,136 

4 

825 

o 

O 

O      - 

+  650 

677,500 

5 

820 

19 

—1672 

—  655-5 

586,847 

6 

825 

8 

—  528 

—  1039-5 

874,225 

7 

330 

0 

—  232 

80,080 

4219 

4-38 

—27 

+2986 

+  2714 

2,200,968 

1,541,452 

—2100 

—  1927 

This  Table  3,  computed  by  the  Prismoidal  Formula  itself,  is  the 
standard  for  all  the  others,  and  gives  the  true  solidities  in  the  section 
of  railroad  under  consideration. 

4th.   Calculation  by  Mean  Proportionals  (or  Geometrical  Average). 
— Professor  Gillespie  says  a  fourth  method,  called  that  of  "Mean 
Proportionals"  is  sometimes,  though  very  improperly,  employed. 
He  gives  the  following  rule  for  Mean  Proportionals. 

Rule. — Add  together  the  areas  of  the  two  ends,  and  a  Mean 
Proportional  between  them  (found   by  extracting   the  Square 
Root  of  their  product)  ;  multiply  the  sum  of  these  three  areas  by 
the  length  of  the  Frustum,  and  divide  the  product  by  three.* 
As  used  by  engineers,  in  working  for  Cubic  Yards  as  the  result, 
tnis  rule  takes  a  somewhat  different  shape,  as  follows: 

Rule. — Multiply  the  sum  of  the  end  areas,  and  the  Square 
Root  of  their  product,  by  the  distance  apart,  and  divide  this  final 
product  by  9  and  by  9. 


*  This   is,   substantially,    Euclid's   Rule  for   the  Frustum    of  a   Pyramid;    Davies' 
Legendre,  VII.  18. 


150  MEASUREMENT  OF  EARTHWORKS. 

The  result  is  always  much  less  than  the  truth  (supposing  the  areas 
taken  between  ground  line  and  road-bed),  for  it  treats  as  Pyramids, 
or  thirds  of  Prisms,  the  wedge-shaped  pieces  which  are  really  halves 
of  Prisms,  and  is  farthest  from  the  truth  when  one  of  the  areas  =  0.* 
So  far.  the  Professor. 

And  this  is  all  correct  when  the  cross-sections  are  limited  between 
road-bed  and  ground  surface  ;  but  if  they  are  extended  to  the  inter- 
section of  the  side-slopes,  or  edge  of  the  diedral  angle  containing  the 
earthwork  solid,  an  entirely  different  state  of  affairs  takes  place,  for  if 
the  road-bed  be  imagined  to  be  gradually  narrowed,  so  that  eventu- 
ally it  vanishes  at  the  intersection  of  the  side-slopes;  then,  at  that 
point,  both  Pyramid  and  Prismoid  coincide,  or  become  equivalent, 
whilst  their  rules  become  correlative  (or  mutually  interchangeable), 
and  either  may  be  used  with  the  same  results  in  point  of  solidity ;  and 
this  is  also  the  case  with  the  "Equivalent  Level  Hights,"  much  used  by 
engineers  since  the  publication  of  Sir  John  Macneill's  work  (London, 
1833),  but  likewise  condemned  by  Professor  Gillespie,  rather  hastily 
as  it  seems  to  the  writer,  and  hardly  upon  sufficient  grounds. 

It  seems  singular  that  this  able  Professor  should  have  overlooked 
the  facts  mentioned  above,  as  he  was  well  acquainted  with  the  method 
of  continuing  calculations  to  junction  of  side-slopes,  including  the 
Grade  Prism  in  the  earlier  stages  of  the  computation,  but  rejecting  it 
at  the  close  (as  may  be  seen  in  his  paper  on  Warped  Solids  (1859)  ). 

Now,  so  long  as  the  cross-section  of  the  earthwork  remains  trape- 
zoidal in  figure,  the  strictures  of  Professor  Gillespie  upon  this  rule 
(commonly  called  the  Geometrical  Average)  are  undoubtedly  correct; 
but  whenever  the  cross-section  becomes  triangular  they  fail  entirely,  as 
also  does  his  similar  censure  on  "  Equivalent  Level  Hights." 

In  evidence  of  this,  we  have  tabulated  (for  ourselves)  the  same 
general  example  as  heretofore  given — both  for  the  Geometrical 

*  Now,  taking  a  case  of  precisely  this  kind  (only  continued  to  intersection  of  slopes) 
— hight  at  one  end  34-5,  at  the  other  0,  with  road-bed  of  30  feet,  slopes  of  2  to  1,  a  length 
of  66  feet,  and  level  on  the  top. 

If  we  compute  this  solid,  either  prismoidally,  or  by  the  usual  rule  for  wedges,  we  have 
for  its  volume  3205  Cubic  Yards  in  round  numbers. 

And  if  we  compute  it  by  Baker's  Rule  (who  treats  such  cases  as  Frusta  of  Pyramids, 
but  with  the  important  addition  of  the  Grade  Prism),  we  find  the  resulting  volume  to  be 
the  same  to  the  nearest  Cubic  Yard. 

For  this  pyramidal  rule  see  Baker's  Earthwork,  London,  1848,  whose  rule  is  similar 
to  that  of  Bidder  and  others,  which  have  always  been  accepted  as  correct  by  English 
engineers,  and  most  certainly  they  are  so. 


CHAP.  VI.— GILLESPIE'S  'RULES,  ETC.— ART.  32.        151 

Average,  and  for  the  Equivalent  Level  Higlits,  merely  carrying  the 
areas  to  the  intersection  of  the  side-slopes,  in  both  cases,  including  at 
first  the  Grade  Prism,  but  excluding  it  after — as  a  common  quantity. 

32 By  these  Tables  we  find  the  solidity  of  Gillespie's  exam- 
ple to  be  precisely  the  same  as  computed  by  him  with  the  Prismoidal 
Formula  (Table  3  above),  and  which  he  has  very  properly  adopted  as 
the  correct  standard  for  all. 

4.  Mean  Proportionals  (or  Geometrical  Average). 

Table  4,  in  illustration  of  computation  by  them,  including  an  entire 
section  of  a  supposed  railroad  =  4219  feet  in  length. 

4.  Road-bed  50 ;  side-slopes  of  excavation  1  £  to  1 ;  of  embank- 
ment 2  to  1. 


Sta. 

Dis- 
tance 
in 
feet. 

To  the 
Road-bed. 

To  intersec- 
tion 
of 
slopes. 

End    Areas   to 
intersection 
of 
slopes. 

Geomet- 
rical 
Mean 
Area. 

Quantities  agreeing 
with  those  of  the 
Prismoidal  Formula. 

Cut. 

+ 

Fill. 

Cut. 

+ 

Fill. 

Sq.  Feet. 

Sq.  Feet. 

Sq.Feet. 

Kxcava. 
Cub.  Feet. 

Embank. 
Cub.  Feet. 

1 
2 
3 
4 
5 
6 
7 

561 
858 
825 
820 
825 
330 

8 

20 

0 

+38 

os°°o 

1«K 

34%: 

31  y, 

20V| 
1'42 

416666 
1802-666 
2(116-666 
416666 

3125 

19845 
840-5 
312-5 

+   816-666 
+  1906-666 
+   916-333 
—    787  5 
—  12915 
—   512-5 

343332 
1,280,136 
577,500 

586847 
874,225 
80,080 

4219 

—27 

+  104% 

—  77 

+  4652-654 

—  3450-0 

+  3639-665 
—  2591-5 

2,200,968 

1,541,152 

In  this  Table  the  Grade  Prism  is  included  at  first,  and  excluded 
afterwards.  Its  sectional  area  is  as  follows : 

Grade  Prism  of  Cut      =  416*666  Square  Feet. 
"  "        Bank  =  312'5          "          " 

To  be  multiplied  for  volume  by  length  of  mass  to  which  it  belongs. 

Altitudes  of  the  Grade  Prism  in  the  Cut  =  161  feet;  on  Bank  = 
mfeet. 

In  computing  quantities  by  Geometrical  Average,  the  following 
generalization  has  occurred  to  the  writer,  which  indeed  may  possibly 
be  a  germ  from  which  the  Prismoidal  Formula  might  have  sprung — 
since  both  the  Arithmetical  and  Geometrical  Means  were  known  in 
the  days  of  Euclid  (  200  B.  c.),  while  the  original  Prismoidal  For- 
mula (so  far  as  we  know)  was  devised  by  Simpson,  as  late  as  A.  D. 
1750. 


152 


MEASUREMENT  OF  EARTHWORKS. 


Thus, 
Double  the  sum  of  End  Areas  -f-  Double  Geom.  Mean 


6 


X  h  =  Solidity. 


Let 

(  A  =  Sum  of  End  Areas.  "I  Then  the  above  f  2  A  +  2  B 
\  B  =  Geometrical  Mean.  /     becomes    .     .-\          6 

Or,  in  its  lowest  terms,  -         -  X  h  =  S,  which  is  the  Geometrical 

o 

Average;  or,  in  substance,  Euclid's  Rule  for  the  Frustum  of  a  Pyra- 
mid ;  and  by  the  aid  of  the  Grade  Prism  strictly  applicable  to  earth- 
works of  a  general  triangular  section  in  ordinary  cases. 

5  Equivalent  Level  Higlits. 
Table  5,  in  illustration  of  computation  by  them. 

5.  Road-bed  50;  side-slopes  of  excavation  1J  to  1;  of  embank- 
ment 2  to  1. 


r 

Sta. 

Dis- 

t'noe 
in 
ft. 

To  the 
Road- 
bed. 

To  intersec- 
tion 
of 

slopes. 

End  Areas  to  inter- 
section of 
slopes. 

Mid.  hta. 
to  inter- 
section 
of  sl'pes. 

Mid-sections,  or 
areas  to  the  inter- 
section of  slopes. 

Quaiititi 
ing  wit 
of  the  P 
Fom 

js  agree- 
h  those 
ismoidal 
ntila. 

Embkt. 

0.    Ft^T. 

Cut. 

+ 

Fill. 

Cut. 

Fill. 

Cut. 

+ 

Fill. 

Feet. 

Sq.   Feet. 

Sq.  Feet. 

Excava. 
C.    Feet". 

1 
2 

3 
4 
5 
6 

7 

561 

858 
825 
820 
825 
330 

O 

18 
20 
O 

+38 

O 
19 
8 

O 

34% 

i<y 

12>£ 
31^ 

20^ 

12% 

416666 
1802-666 
2016-666 
416-666 

3125 

1984-5 
8405 
312-5 

+  25-666 
-r  35-666 
+  26  666 
—  22000 
—  26-000 
—  16  500 

988166 
1908-166 
1066  666 

968-0 
1352  0 
544-5 

343,332 
1.280,136 

577,500 

586,847 
874,225 
80,080 

4219 

—27 

+  104% 

—77 

+  4652-654 

—  3450-000 

+  87-998 
—  64-500 

+  3962  998 

—  2864-5 

2,200;968  1,541,152 

In  this  Table  the  Grade  Prism  is  included  in  the  earlier  operations, 
and  excluded  in  the  later  ones.  Its  sectional  area  is  as  follows: 

Grade  Prism  of  Cut  =  416'66  Square  Feet. 
Fill  =  312-50       " 

To  be  multiplied  for  volume  by  the  length  of  mass  to  which  it 
belongs. 

Altitudes  of  the  Grade  Prism  in  the  Cut  =  16f  feet;  on  Bank  = 
12*  feet. 

33.  From  the  preceding  discussion  in  the  present  chapter  we  are 
justified  in  declaring  that  all  the  following  rules  and  formulas 
(detailed  above)  are  equivalent  in  their  results  for  volume — when  pro- 


CHAP.  VI.-GILLESPIE'S  RULES,  ETC.— ART.  33.        153 

perly  corrected  and  appropriately  used ;  and  that  they  all  give  the 
same  solidity  in  the  end  as  No.  3  does,  which  is  the  standard  for  ALL. 

1.  Arithmetical  Average  to  Road-bed  (with  correction). 

2.  Middle  Areas  to  Road-bed  (with  correction). 

3.  Prismoidal  Formula  (the  standard  for  all)  to  Road-bed,  or  to 

the  intersection  of  slopes — either. 

4.  Geometrical  Average  to  intersection  of  slopes. 

5.  Equivalent  Level  Hights  to  intersection  of  slopes. 

All  these  are  fully  described  above,  and  the  tabular  statements 
bearing  the  same  number  show  in  each  case  the  results  of  the  calcu- 
lations for  volume,  agreeing  uniformly  with  the  computations  for 
solidity,  made  by  means  of  the  Prismoidal  formula. 

In  concluding  his  notices  of  the  method  of  computing  the  contents 
of  earthworks,  by  means  of  the  Prismoidal  Formula,  Professor  Gilles- 
pie  gives  some  special  rules,  transformed  from  it,  which  are  doubtless 
valuable  in  certain  cases,  but  do  not  appear  to  be  of  general  applica- 
tion; he  also  gives  formulas  for  a  series  of  equal  distances  apart  sta- 
tions, such  as  are  usually  found  in  the  location  of  railroads. 

These  are  intended  to  be  applied  to  a  central  core,  or  body  of  the 
work,  based  upon  the  road-bed,  to  be  calculated  by  itself,  and  then 
the  slopes,  to  be  computed  separately  or  together,  and  added  in  with 
the  core,  so  as  to  form  finally  the  volume  of  the  ivhole  prismoidal  mass. 

This  idea  of  separating  the  core  or  body  from  the  slopes,  calculating 
them  independently,  and  adding  them  together,  seems  to  have  occurred 
to  a  great  many  .engineers,*  and  forms  the  theme  of  nearly  a  dozen 
books  on  the  subject  of  Earthwork  Measurements — here  or  abroad. 

Indeed,  the  very  first  special  work  on  the  mensuration  of  earth- 
works, which  was  published  in  this  country — that  of  E.  F.  Johnson, 
C.  E.  (New  York,  1840),  adopted  this  system,  and  furnished  a  series  of 
Tables  to  facilitate  its  operation ; — it  was,  however,  briefly  explained 
before,  in  Lieut.-Col.  Long's  valuable  Railroad  Manual  (Balti- 
more, 1828),  which  was  the  first  to  treat  the  subject  in  this  country, 
and  was,  in  fact,  the  pioneer  of  technical  railroad  literature  in  the 
UNITED  STATES. 

Nevertheless,  the  method  of  Core  and  Slopes  has  never  come  into 
general  use,  though  often  revived  from  time  to  time  by  new  writers, 
apparently  unacquainted  with  the  literature  of  this  subject. 

*  Amongst  others,  it  is  the  method  of  Bidder,  who  followed  Macneill  in  the  earlier 
days  of  English  railroads. 


154 


MEASUREMENT  OF  EARTHWORKS. 


34.  .  .  1  .  .  Comparison  of  Gillespie's  Main  Example  and  the  Method^ 
of  Roots  and  Squares. 

Professor  Gillespie's  chief  example,  of  a  heavy  Cut  and  Fill,  form- 
ing an  entire  section  of  railroad,  4219  feet  long,  must  by  this  time 
be  so  familiar  to  engineers,  and  others,  in  consequence  of  the  exten- 
sive circulation  of  his  Manual  of  Roads  and  Railroads,  since  its  origi- 
nal publication  in  1847,  that  we  have  selected  it  as  the  most  suitable, 
or  at  least  the  best  known*  for  the  purpose  of  comparison  with  our 
Third  Method  of  Computation — that  by  Roots  and  Squares. 

We  therefore  give  a  Table  No.  6  (below),  which  contains  in  the 
first  5  columns  the  data  given  by  Professor  Gillespie,  and  in  the  last 
6  the  results  of  the  computation  by  Roots  and  Squares,  which  will  be 
found  to  agree  exactly  with  those  obtained  above,  by  means  of  the 
Prismoidal  Formula — accepted  as  being  a  correct  standard  for  com- 
parison. 

6.   Comparison  of  Example,  with  Hoots  and  Squares. 

Including  (as  before)  an  entire  section  of  a  supposed  railroad  — 
4219  feet  in  length. 

6.  Road-bed  50;  side-slopes  of  excavation  II  to  1 ;  of  embank- 
ment 2  to  1. 


Sta. 

Dis- 
tance 
in 
feet. 

End 
Areas 
in 
Sq.  Tt. 

Centre 
Rights 
in  feet. 

End 
Areas 
increased 
by  Grade 
Triangle. 

Square 
Roots 
of 
End 
Areas. 

Suma 
of 
Square 

Roots. 

Squares  of 
sums,  or  4 
times  the 
mid- 
section. 

Quantities  agree- 
ing with  those 
given  by  the 
Prismoidal 
Formula. 

Cut  + 
Fill  — 

Cut 

+ 

Fill 

O 

19 

8 
O 

Sq.  Feet. 

Feet. 

Feet. 

Feet. 

C.   Feet. 

0.   Feet. 

1 
2 
3 

4 

5 
6 

7 

561 

858 

825 

820 
825 
330 

o 

+1386 
-j-1600 

0 
—1672 
—  528 

0 

g 

20 
0 

+38 

+  416% 
+1802% 
+2016% 
<  +  416% 
)—  312J>£ 
—  1984U 
—  840^ 
—  312}£ 

+    20-42 
+    42-46 
+   44-91 
+    20-42 
—    17-68 
—    44-55 
—    28-99 
—    17-68 

62-88 
87-37 
65-33 

—   62-23 
—   73-54 
—   46-67 

3954 
7634 

4268 

—    3872 

—    5408 
—    2178 

343,332 

1,280,136 

577,500 

586,847 
874,225 
80,080 

4219 

+2986 
—2200 

—27 

+4652% 
—3450 

+  128-21 
—  108-90 

215-58 
—  182-44 

15856 
—  11458 

2,200,968 

1,541,152 

In  the  above  Table  (as  in  the  others),  the  cross-sections — in  the 
data  given — being  level  trapezoids  from  ground  to  road-bed,  we  neces- 

*  Besides,  this  example,  originated  by  F.  W.  Simms,  C.  E.  (London,  1836),  has  been 
before  the  public  for  many  years,  having  been  first  published  in  our  country  in  Alexan- 
der's edition  of  Simms  on  Levelling  (Baltimore,  1837) ;  from  which,  or  the  original,  it  was 
copied  by  Professor  Gillespie.  In  the  work  above  mentioned,  Mr.  Alexander  gives  every 
detail  of  the  computation  of  this  example,  by  the  Prismoidal  Formula,  at  great  length, 
and  so  indeed  does  Simms. 


CHAP.  VI.—  GILLESPIE'S  RULES,  ETC.—  ART.  34.        155 

sarily  add  in  this  mode  of  computation  (to  intersection  of  slopes)  the 
Grade  Triangle,  and  deduct  it  again  near  the  close  of  the  operation. 

Road-bed  50  ;  side-slopes  of  excavation  =  1  J  to  1  ;  of  embank- 
ment =  2  to  1. 

Grade  Triangle  of  Cut,  area  =  416f  Sq.  Ft.  —  altitude  =  16$  Feet. 
"     Fill,     "     =  312£   "     "    —      "         =  12*     " 


Where  the  distances  apart  stations  are  uniform  in  length  and  even 
in  number,  the  method  of  Roots  and  Squares  enables  us  to  employ  a 
very  simple  modification  of  Simpson's  Multipliers,  as  has  been  already 
shown  in  Chapter  IV.,  so  as  to  compute  with  ease  and  expedition  an 
entire  cut  or  fill,  at  a  single  operation,  or  one  station  only,  at  pleasure. 


CHAPTER  VII. 

PRELIMINARY  OR  HASTY  ESTIMATES,  COMPUTED  BY  SIMPSON'S  RULE 

FOR  CUBATURE. 

35 Preliminary,  and  often  hasty  estimates  of  earthworks, 

are  constantly  required  by  engineers  prior  to  deciding  upon  railroad 
routes,  or  their  modifications,  and  indeed  are  generally  necessary  in 
determining  the  relative  merits  of  engineering  lines — (amongst  which 
there  are  always  alternatives} — since  few  can  undertake  to  settle  pro- 
perly any  important  questions  relating  to  their  comparative  value, 
without  some  serious  consideration,  for  which  the  Preliminary  Esti- 
mates, on  various  lines  surveyed,  supply  a  proximate  foundation,  by 
aiding  without  controlling  the  judgment  of  the  engineer. 

Exploring  Lines,  preparatory  to  the  final  location  of  a  railway,  are 
indispensable,  and  in  a  difficult  country  may  extend  to  tenfold  the 
length  of the  final  line,  while  the  time  allowed  to  engineers  being  usually 
extremely  short,  the  estimates  of  quantities  on  these  Preliminary  Sur- 
veys are  necessarily  hasty,  and  consequently  imperfect — but  neverthe- 
less demand  rapidity  in  execution,  however  made. 

For  this  there  seems  to  be  no  remedy ;  all  we  can  do  is  to  endeavor 
to  point  out  a  method  for  hasty  estimates,  more  correct  and  more  expe- 
ditious than  those  usually  employed,  and  to  this  we  shall  confine  our- 
selves in  the  present  chapter. 

Exploring  lines  are  usually  traced  with  stations  at  double  distance, 
or  200  feet  apart — and,  indeed,  sometimes  on  plain  ground  the  dis- 
tance apart  stations  has  been  stretched  (to  save  time)  as  far  as  400 
or  600  feet ; — and  as  this  last  distance  is  about  the  longest  range 
which  gives  distinct  vision  for  the  Engineer  Levels  in  use  in  this 
country,  it  ought  rarely  to  be  exceeded,  as  a  general  rule;  while  at 
least,  the  distance  of  200  feet  apart  stations,  or  double  distance  of  loca- 
156 


CHAP.  VII.— HASTY  ESTIMATES.— ART.  35.  157 

tion,  furnishes  good  information  of  the  ground,  and  also  enables  the 
exploring  party  to  proceed  rapidly  enough  to  gain  an  adequate  know- 
ledge of  the  country,  without  much  loss  of  time. 

Nevertheless,  the  rules  we  suggest  will  apply  to  any  uniform  dis- 
tance apart  stations  of  exploring  line,  which  may  be  deemed  advisable 
by  the  engineer  in  charge:  but  the  longer  the  distance  between  sta- 
tions, the  less  accurate  will  be  the  estimate  in  general. 

We  propose  to  apply  Simpson's  celebrated  rule  for  cubature  (the 
accuracy  of  which  is  well  known)  to  Preliminary  or  Hasty  Estimates, 
taking  as  data  the  centre  hights  and  surface  slopes  alone;  the  former 
to  the  nearest  foot  of  hight  or  depth,  from  ground  to  intersection  of 
side-slopes,  and  the  latter  to  the  nearest  5°  of  average  ground  slope 
across  the  line,  leaving  special  cases  to  be  dealt  with  by  the  engineer, 
according  to  rules  of  his  own. 

We  have  provided  proximate  tables  (very  nearly  correct)  to  facili- 
tate these  hasty  operations,  and  would  also  suggest  that,  in  all  cases 
of  Preliminary  Estimates,  the  resulting  quantities  of  earthwork  should 
be  augmented  ten  per  cent.: — this  addition  will  give  full  quantities, 
and  has  been  shown  by  long  experience  to  be  ample  to  meet  the  usual 
contingencies  which  always  arise  in  the  construction,  and  cannot  be 
foreseen,  and  of  which,  in  fact,  it  must  be  confessed,  the  engineer  in 
charge  (often  unknown  to  himself)  almost  invariably  takes  the  most 
favorable  view',  and  hence  the  greater  necessity  exists  for  some  appro- 
priate allowance  beyond  the  net  result  of  the  calculations. 

Simpson's  Rule  for  Cubature,  using  cross-sections  instead  of  ordi- 
nates  (as  we  have  before  shown),  is  as  follows: 

— i — --Jt X  D  =  Solidity. 

o 

(  Sometimes  2  D,  and  6  for  divisor,  are  used,  and  are  equivalent.) 

A  =  Sum  of  extreme  end  ordinates,  or  sections. 
B  =  Sum  of  cross-sections  standing  on  even  numbers. 
C  =  Sum  of      "         "          "         "       odd  numbers. 
D  =  The  common  interval,  or  distance  apart  sections. 

Simpson's  rule  above  is  limited  to  an  even  number  of  equal  spaces. 


158  MEASUREMENT  OF  EARTHWORKS. 

And  it  must  be  observed  that  in  its  application  it  is  always  best  to 
prepare  a  rough  profile  of  the  line  run,  and  under  the  regular  num- 
bers to  pencil  forward,  from  the  beginning  of  the  cut  or  fill  to  be 
computed,  the  series  of  numbers  1,  2,  3,  4,  etc.  No.  1  always  stand- 
ing at  the  place  of  beginning ;  it  is  this  series  of  numbers,  so  arranged, 
which  are  referred  to  in  the  rule  above  as  even  and  odd. 

By  this  rule  it  is  best  to  compute  entire  and  separately  each  cut  and 
each  fill  encountered  by  the  line  ;  and  if  the  whole  number  of  equal 
intervals  or  stations,  in  any  cut  or  fill,  should  be  an  odd  number,  then 
one  station  of  the  common  length,  at  beginning  or  end  (or  indeed  any 
where  deemed  most  suitable),  should  be  struck  off  temporarily,  and 
reserved  for  separate  calculation  ;  while  the  body  of  the  work  thus 
reduced,  to  an  even  number  of  common  intervals,  comes  directly  within 
the  rule,  and  can  be  calculated  as  a  whole,  while  the  detached  sta- 
tion, computed  by  itself,  may  be  added  in  near  the  close  of  the  ope- 
ration. 

It  will  always  be  found  briefer  and  better  in  using  this  and  similar 
rules,  to  aim  first  at  finding  a  General  Mean  Area,  which,  multiplied 
by  the  proper  length  or  distance,  will  give  the  solidity ;  but  it  is  still 
better,  having  the  General  Mean  Area  in  square  feet,  to  use  our  Table 
at  the  end  when  the  result  is  desired  in  Cubic  Yards. 

36 Instead  of  employing  Simpson's  Formula,  as  it  stands 

above,  it  will  be  often  more  convenient  to  use  the  multipliers  which 
represent  it — these  are  known  as  Simpson's  Multipliers*  and  are  as 
follows : 


For  two  equal  int6rvals,  apart  sections,  Mults. 


(  Divisors  6;  qiiot?ent,Mean 
I,  4,  l.<    Areas ;  factors  for  length 
(  =  double  interval. 


"  four"  "  "  "  "  =  1, 4, 2, 4, 1.  f     Divisors    3;    quotient, 

««  six      «'  "  "  "  "  =  1.  4,  2,  4,  2,  4,  1.  J  Mean   Areas ;  factors   for 

«  eiaM  "  "  "  "  "  =          1,  4,  2,  4,  2,  4,  2,  4, 1. 1  length    =   single    inter- 

«  ten      "  "  "  "  "  =  1,  4,  2,  4,  2,  4,  2,  4,  2,  4, 1.  [veil. 

The  first  set  of  multipliers,  their  divisors,  and  factors  for  length,  are 
clearly  those  of  the  Prismoidal  Formula,  which  evidently  forms  the 
basis  of  this  famous  rule. 

Indeed,  it  is  easy  to  show  by  diagrams  how  this  rule  may  probably 
have  been  formed,  by  the  eminent  mathematician,  with  whom  it 
originated,  about  the  year  1750  ;  and  also  how  intimately  it  appears 
to  be  connected  with  the  Prismoidal  Formula. 

*  Rankine's  Useful  Rules  and  Tables,  2d  edition,  London,  1867,  page  64. 


CHAP.  VII.— HASTY  ESTIMATES.— ART.  36. 


159 


See  Figs.  11  and  78,  following. 

Suppose  Figs.  11  and  78  to  represent  front  views  of  four  planes,  A, 
B,  C,  D,  or  of  four  solids  with  a  thickness  of  unity,  all  standing  on 
the  level  base  line  EF,  and  that  their  respective  ordinates,  or  cross- 
sections  (correllative  in  Simpson's  Rule  for  Cubature),  are  dimen- 
sioned as  marked  upon  the  figures. 


Tig.  77; 


10,     10.   I  10.      10.      jtO.  I  1O.     1O.       10.       30 


31O 


Kg.  73. 


1.  Suppose  the  solids  to  be  separated  from  each  other  by  the  dis- 
tance of  10  feet  (or  any  other),  and  let  each  be  computed 
independently  by  means  of  Simpson's  Multipliers,  or  as  they 
are  all  exactly  alike,  let  one  be  computed  and  multiplied  by 
4,  as  follows : 


This  is  clearly 

a 
Prismoidal  Compulation. 


Cross-sees.    Simpson's        Results  in 
in  Sq.  Ft.        Mults.  Sq.  Ft. 


X  1 
X  4 
X  1 


=       1 

=»     16 


6)18 
Mean  Area  =       3  X  20 


60  A. 


60  X  4  =  240  Cubic  Feet  = 


160  MEASUREMENT  OF  EARTHWORKS. 

2.  Now,  suppose  the  solids  to  be  slid  along  the  base  line  EF, 
until  they  come  in  actual  contact  with  each  other,  as  shown 
in  Fig.  78.  Then  it  becomes  evident  that  the  intermediate 
sections  at  odd  numbers  (1,  3,  etc.),  which,  in  the  detached 
solids,  Fig.  77,  were  used  but  once,  are  here,  when  combined, 
to  be  used  twice;  while  the  mid-sections,  or  those  at  even 
numbers,  are  to  be  used  four  times,  and  the  extreme  end  sec- 
tions only  once  each ;  so  that  they  become,  in  effect,  when 
treated  thus,  the  Multipliers  of  Simpson ;  while  the  divisor 
is  changed  to  3,  because  the  common  interval  is  reduced 
one-half; — and  the  volume  of  the  four  solids,  when  aggre- 
gated together,  so  as  to  form  a  single  body,  would  be  com- 
puted by  Simpson's  Rule,  or  by  his  Multipliers,  as  follows: 


By  Sim 

Common 
9    4-     64    -4-    6       Interval. 
„,/,«»«  Rut*                                 V   10 

240,  o.s  above. 

s.            Sq.  Ft. 
-j 

=     16 

=       2 

Sees.           Multi 

/I     X     1 
|4     X     4 
1X2 

\4     X 

4 

=     16 

By 

Simpson's  Multipliers,  1  1     X 

2 

=       2 

with  8  equal  intervals.          \  4     X 

4 

=     16  , 

(1      X 

=       2 

4     X 

4 

=     16 

1     X 

1 

=      1 

3)72 

General  Mean  Area    . 

t 

=     24 

Common  Interval  .     . 

. 

=      10 

Result  same  as  before  . 

. 

=     240  C. 

Feet. 

As  Simpson's  Rule  is  an  important  one,  we  hope  the  above  digres- 
sion to  explain  it  fully,  and  the  foundation  on  which  it  rests,  will  be 
excused  by  the  reader. 

37.  Having  then  taken  off  from  a  rough  profile  of  the  line  run  the 
centre  bights  to  the  nearest  foot,  and  from  the  field  notes  ascertained 
the  average  surface  slope  at  each  station  to  the  nearest  5°,  we  enter 
Tables  2,  3,  and  4,  and  obtain  the  triangular  areas  to  the  intersection 
of  the  side-slopes  (supposed  to  be  prolonged  to  meet),  to  the  nearest 
foot  of  area,  for  rock  cutting,  earth  cutting,  or  embankment — each  of 


CHAP.  VIL— HASTY  ESTIMATES.— ART.  37.  161 

these,  that  we  may  require,  we  set  down  separately  in  a  column,  and 
where  a  case  occurs  of  a  hight  exceeding  the  limits  of  the  Tables 
named,  then  we  resort  to  the  initial  triangles  of  Table  1,  by  means  of 
which  the  area  due  to  any  hight  whatever  may  easily  be  ascertained ; 
then,  if  we  find  we  have  an  even  number  of  equal  stations,  we  apply 
Simpson's  Multipliers  to  the  column  of  areas,  and  speedily  compute 
the  solidity. 

But  if  the  equal  intervals  or  stations  are  found  to  be  uneven  in 
number,  strike  off  one  station  temporarily  for  independent  calcula- 
tion, and  then  the  number  of  intervals  becoming  erent  we  are  ready 
to  apply  Simpson's  Multipliers,  in  a  column  parallel  to  that  of  areas, 
and  beginning  at  1,  as  1,  4,  2,  4,  2,  4,  etc.,  multiplying  each  cross-sec- 
tion by  its  proper  factor,  and  placing  the  results  in  a  third  parallel 
column,  which  we  sum  up  and  divide  the  total  by  3  (giving  a  Mean 
Area  as  the  quotient),  add  to  this  the  mean  area  of  the  station 
reserved  (if  any),  which  gives  a  General  Mean  Area,  to  be  multiplied 
by  the  equal  interval,  or  length  of  station — say  200  feet,  or  whatever 
distance  has  been  adopted  and  used  as  a  common  interval  or  station 
— the  result  will  be  cubic  feet,  from  which  cubic  yards  (if  desired) 
can  easily  be  found. 

But,  inasmuch  as  the  quotient  of  3  (with  the  mean  area  of  the 
reserved  station  (if  any)  added  in)  is  a  General  Mean  Area — usually 
in  square  feet — it  will  be  found  more  convenient,  and  usually  more 
accurate,  to  use  it  in  connection  with  our  Table  5,  at  the  end  of  the 
Book,  to  find  the  cubic  yards  which  may  be  desired,  according  to  the 
directions  preceding  the  Table. 

We  will  now  proceed  to  give  examples  of  the  process  above 
explained,  and  for  this  purpose  we  will  take  the  adjacent  bank  and 
rock  cut,  profiled  on  Fiy.  76,  Art.  24,  as  being  an  appropriate  exam- 
ple of  this  expeditious  method  of  computing  an  embankment,  or  an 
excavation  in  a  single  body,  with  sufficient  accuracy  for  the  purpose 
contemplated,  and  without  unusual  delay. 

Fig.  76.  BANK. 

Here  we  find  the  Bank  to  be  1000  feet  in  length  between  the  grade 
points,  or  5  intervals  of  200  feet  each ;  the  number  of  intervals  being 
uneven,  we  must  temporarily  omit  one  station  to  bring  this  case  within 
the  rule ;  let  the  station  omitted,  and  to  be  calculated  independently, 
be  from  5  to  7  =  200  feet. 
11 


162 


MEASUREMENT  OF  EARTHWORKS. 


Tabulation. 


Sta.       Areas. 
1 

3 


Mults. 


24  X  1  = 

495  X  4  « 

5  and  7     3123  X  2  = 
united. 

9     1197  X  4  = 

11         24  X  1  = 


Sq.  Feet. 

24 

1980 
6246 

4788 
24 


3)13062 
4354 


Partial  Mean  Area. 


Add  area  of  reserved  station. 

The  hight  of  the  embank- 
ment and  the  surface-slope  at 
5  and  7  being  the  same,  this 
reserved  station  is  a  Prism,  of 
which  the  base,  or  sectional 
area,  is  3123  square  feet,  and 
length  =  200  feet  .... 

General  Mean  Area.     .     . 


Solidity     .-..«..;  •>'-•-.-; 

Or, ;;   .: 

Tabulated,    by    Roots     and 
Squares,  in  100  feet  stations     . 

Difference  about  the  half  of 
one  per  cent,  more 


=  »  3123      =  Mean  Area,  reserved 

station. 
=     7477  Square  Feet. 

200       Common  Interval. 

=  1495400       Cubic  Feet. 
=       55385        Cubic  Yards. 

=       55088 

=      -f297          "        " 


Tabulated  by  Roots  and  Squares  in  100  feet  stations,  as  though  for 
a  final  estimate,  the  Bank  in  our  example  contains  55,088  Cubic 
Yards,  while  by  our  hasty  process  the  result  is  55,385  Cubic  Yards, 
or  297  Cubic  Yards  more.  As  this  difference  is  but  little  more  than  the 
half  of  one  per  cent,  upon  the  true  amount,  it  can  hardly  be  consid- 
ered as  excessive  for  a  method  as  brief  and  simple  as  that  under  con- 
sideration here. 

Fig.  76.  ROCK-CUT. 

The  Rock-Cut,  like  the  Bank  connected  with  it,  and  tabulated 
above,  is  1000  feet  in  length  between  the  grade  points,  or  5  intervals 
of  200  feet  each,  which,  being  an  uneven  number,  we  must  tempora- 


CHAP.  VII.-HASTY  ESTIMATES.— ART.  38. 


163 


rily  omit  one  station,  and  calculate  it  separately,  to  make  the  number 
of  intervals  even,  and  bring  it  within  the  scope  of  Simpson's  Rule. 
Let  the  station  reserved  be  from  19  to  21  =  200  feet. 


Tabulation. 


Sta.    Areas.  Mults. 

11     192     X     1 
13     646     X     4 


15     975 


17     589     X     4 
19     771     X     1 


Station  reserved  from  19  to 
21,  to  make  the  number  of  in- 
tervals even,  as  required  by 
the  Rule  of  Simpson. 

19  =  771  X  1  =    771 

20  =  433  X  4  =  1732 

21  =  192  X  1  =    192 

6)2695 

Mean  Area  =    449 
General  Mean  Area 


Solidity    .     .     .     . 

Tabulated     by     Roots    and 

Squares,  in  stations  of  100  feet 

Diff.  about  1£  per  cent,  less 


Sq.  Feet. 

=     192 

=  2584 

=  1950 

=  2356 

=     771 

3)7853 

2618 


Partial  Mean  Area, 


449 


3067 
200 


Mean  Area,  reserved 
station. 

Square  Feet. 
Common  Interval. 


=  613400  =  22718  Cubic  Yards. 

=  623298  =  23085      " 
=       9898  =  —367      " 


38 It  will  be  observed  that  in  the  preceding  computations 

the  Grade  Prism  is  not  taken  into  the  account,  as  it  is  deductive  on 
both  sides,  and  the  only  object  in  hand  is  a  comparison. 

The  triangular  section,  or  area  of  the  Grade  Prism,  is  the  minimum 
area  found,  in  the  methods  of  computation  which  go  down  to  the 
junction  of  the  side-slopes,  and  always  occurs  when  the  road-bed 
comes  to  grade,  or  the  level  hight  on  the  centre  line  is  0. 

And  we  repeat,  it  is  necessary  to  be  careful  that  the  volume  of  the 
Grade  Prism  (always  included  in  the  earlier  steps  of  such  calcula- 
tions) is  duly  deducted  before  the  close  of  the  operation,  in  order  to 
determine  the  solidity  above  the  road-bed  in  cutting,  or  below  it  in 
filling. 


164  MEASUREMENT   OF  EARTHWORKS. 

We  may  here  add  that  the  earth  cutting  profiled  ante,  and 
there  correctly  computed  by  Roots  and  Squares,  if  calculated 
with  Simpson's  Multipliers  by  the  hasty  process  above  given,  in  sta- 
tions of  200  feet,  as  though  it  were  part  of  an  exploring  line,  would 
give  as  follows : 

Volume  of  Grade  Prism  omitted  in  both. 

C.  Yards. 

/Tabulated  ante,  in  100  feet  stations ==  18684 
"        by  our  Hasty  Process,  200  feet  stations  .     .     .  =  18378 


i 


Difference  about  H  per  cent,  less =      306 

So  that  this  brief  and  hasty  process,  being  very  expeditious  and 
proximately  correct  (usually  varying  only  1  or  2  per  cent,  from  the 
truth),  may  be  safely  accepted  as  adequate  for  the  determination  of 
the  quantities  of  earthwork,  which  may  be  needed  in  rough  estimates, 
or  for  the  comparison  of  exploring  lines. 

For  the  purpose  of  furnishing  additional  aid  in  expediting  Prelimi- 
nary Estimates,  we  annex  four  small  Tables,  which  will  be  found 
quite  convenient. 


TABLES.  165 

"LIBRA  u  v 

UNIVERSITY   OJ 


TABLES 


CALIFORNIA. 


1,  2,  3,  and  4. 

For  use  in  Hasty  or  Preliminary  Estimates. 

Viz:  1.  Initial  Triangles  to  a  hight  of  unity,  and  various  side  and 
surface  slopes. 

Triangular  Areas  to  Intersection  of  Slopes. 

Side-slopes.  Surface-slopes. 

2.  Kock  Cut  i  to  1,  and  0°,    5°,    10°,     15°,    20°. 

3.  Earth  Cut  1  to  1,  and   "        "         "          " 

4.  Embankment        and   " 

In  using  Tables  2,  3,  and  4,  the  centre  hight  is  generally  to  be 
taken  to  the  nearest  foot  (though  tenths  might  be  used),  and  the 
ground  surface  slope  to  the  nearest  5° — these  being  thought  sufficient 
for  rough  estimates — and  if  the  centre  hight  should  exceed  the  limits 
of  the  Tables,  then,  by  using  the  Initial  Triangles  of  Table  1,  the  area 
of  the  cross-section  for  any  hight  whatever  can  be  easily  ascertained. 
If  the  centre  hights  necessarily  contain  tenths  of  feet,  they  may  be 
proportioned  for  by  the  columns  in  the  Tables  for  that  purpose. 

Note. — All  the  triangular  areas  in  Tables  2,  3,  and  4,  extend  from 
ground  line  to  junction  of  side-slopes  prolonged,  or  edge  of  the  diedral 
angle,  which,  with  ground  surface,  bounds  on  every  side  the  earth  work 
solid.  The  road-bed,  or  grade  line,  may  be  assumed  to  cross  the  tri- 
angle at  any  given  distance  from  the  angle  of  intersection ;  but  the 
volume  of  the  Grade  Prism  must  always  be  ascertained  and  deducted 
at  the  close  of  the  operation,  in  every  calculation  involving  the  trian- 
gular areas  of  the  Tables.  The  altitude  of  the  Grade  Triangle  is 
invariably  =  road-bed  -f-  2  r,  and  its  area  will  be  found  opposite  to 
this  hierht  in  the  0  column  of  the  Tables. 


166 


MEASUREMENT  OF  EARTHWORKS. 


TABLE  1. 

Initial  Triangles,  to  a  hight  of  unity,  with  side-slopes  of  i  to  1  for 
Rock;  1  to  1  for  earth;  1&  to  1  for  embankment;  and  ground  sur- 
face slopes  of  0°,  5°,  10°,  15°,  20°.  All  computed  to  six  places  of 
decimals,  and  all  extending  from  ground  line  to  intersection  of  side- 
slopes. 


Side-slopes. 

Ground   Surface-elopes. 

Ratio. 

Angle. 

Cot. 

Tan. 

0° 

5° 

10° 

15° 

20° 

of  Trian.  Tables. 

Tan. 
=  •() 

Tan. 

=  -0875 

Tan. 
=  -1763 

Tan. 

=  -2679 

Tan. 

=  •3640 

Y3to  1 
1      to  1 
1^  to  1 

71°  34' 
45° 
33°  41' 

0-3333 
1 
1-5 

3 
1 
•6666 

0'333333 
1-5 

0-333586 
1-007713 
1-526688 

0-334457 
1-032088 
1-613298 

0-335682 
1-077350 
1-790002 

0-338280 
1-152663 
2-137798 

Note. — A  similar  Table  may  easily  be  extended  to  any  other  side, 
or  surface-slope,  and  such  extension  would  often  be  found  useful  to 
the  engineer. 

Application  of  the  above  Table. 

Hide. — For  any  given  hight,  to  find  the  triangular  area,  when  con- 
ditioned as  above. 

Multiply  the  Square  of  the  Given  Hight  by  the  Tabular  Area  of  the 
Initial  Triangle. 

Example. 

Let  the  given  hight  be  26'4  feet,  the  side-slope  1  to  1,  and  the 
ground  surface-slope  20°. 

Then,  (26'4)2  X  1*152663  =  803*36  square  feet  =  area  of  triangle 
required. 


TABLES. 


167 


Triangular  Areas,  in  square  feet,  for  side-slopes  of  i  to  1,  to  intersec- 
tion of  slopes,     (r  =  i)     Slope  angle  =  71°  34'. 


TABLE  2—Ri>ck-cirt. 


Hight 

iii 

Surf.-slope  O°. 

Surf.-elope  5°. 

Surf.-elope  10°. 

Surf.-slope  15°. 

Surf.-slope  »  0°. 

Hight 
in 

feet. 

Pro. 

Pro. 

Pro. 

Pro. 

Pro. 

feet. 

Areas, 

for-1. 

Areas. 

for  -1. 

Areas. 

for  -1. 

Areas. 

for  -1. 

Areas. 

for  -1. 

1 

•3333 

•03 

•3336 

•03 

•3346 

•03 

•3357 

•03 

•3383 

•03          1 

2 

1-3333 

•10 

1-3 

•10 

1-3 

•10 

13 

•10 

14 

•10 

2 

3 

3 

•17 

3 

•17 

3 

•17 

3 

•17 

3 

•17 

3 

4 

5-3333 

•23 

5 

•23 

5 

•23 

5 

•23 

5 

•24 

4 

5 

83333 

•30 

8 

•30 

8 

•30 

8 

•30 

8 

•30 

5 

6 

12 

•37 

12 

•37 

12 

•37 

12 

•37 

12 

•37 

6 

7 

163333 

•43 

16 

•43 

16 

•43 

16 

•44 

17 

•44 

7 

8 

21-3333 

•50 

21 

•50 

21 

•to 

22 

•50 

22 

•51 

8 

9 

27 

•57 

27 

•57 

27 

•57 

27 

•57 

28 

•58 

9 

10 

33-3333 

•63 

33 

•63 

33 

•64 

34 

•64 

34 

•64 

10 

11 

4O3333 

•70 

40 

•70 

41 

•70 

41 

•71 

41 

•71 

11 

12 

48 

•77 

48 

•77 

48 

•77 

48 

•77 

49 

•78 

12 

13 

56-3333 

•83 

56 

•83 

57 

•84 

57 

•84 

67 

•85 

13 

14 

65-3333 

•90 

65 

•90 

66 

•90 

66 

•91 

66 

•91 

14 

15 

75 

•97 

75 

•97 

75 

•97 

76 

•98 

76 

•98 

15 

16 

85-3333 

1-03 

85 

1-03 

86 

1-04 

86 

1-04 

87 

1-05 

16 

17 

96-3333 

1-10 

96 

1-10 

97 

1-10 

97 

1-11 

98 

I'll 

17 

18 

K>8 

1-17 

108 

1-17 

108 

1-17 

109 

1-18 

110 

1-18 

18 

19 

120-333.'} 

1-23 

121 

1-23 

121 

1-24 

121 

124 

122 

1-25 

19 

20 

133-3333 

1-30 

133 

1-30 

134 

1-30 

l;» 

1-30 

135 

131 

20 

21 

147 

1-37 

147 

1-37 

148 

1-37 

148 

1-37 

149 

1-38 

21 

161-3333 

1-43 

161 

1-43 

162 

1-44 

163 

1-44 

164 

1-45 

22 

23 

176-3333 

1-50 

176 

1-50 

177 

1-50 

178 

1-51 

179 

1-52 

23 

24 

192 

1-57 

192 

1-57 

193 

157 

194 

1-68 

195 

1-59 

24 

25 

2  >8-3333 

1-63 

209 

1-63 

209 

164 

210 

1-64 

212 

1-66 

25 

26 

225-3333 

1-70 

226 

1-70 

226 

V70 

227 

1-71 

229 

1-72 

26 

27 

243 

177 

243 

1-77 

244 

1-77 

245 

1-78 

247 

1-79 

27 

23 

261-3333 

1-83 

262 

1-84 

262 

1-84 

263 

1-85 

265 

1-86 

28 

29 

280-3333 

1-9!) 

281 

1-90 

281 

1-91 

282 

1-91 

285 

1-93 

29 

31) 

300 

1-97 

300 

1-97 

301 

1-97 

302 

1-98 

305 

2-00 

30 

31 

320-3333 

2-03 

321 

2-04 

322 

2-04 

323 

2-05 

325 

2-06 

31 

32 

341-3333 

2-10 

342 

2-10 

343 

2-11 

344 

2-12 

346 

2-13 

32 

33 

363 

2-17 

363 

2-17 

36i 

2-17 

366 

2-18 

308 

2-20 

33 

34 

385-3333 

2-23 

386 

2-24 

387 

224 

388 

225 

391 

2-27 

34 

35 

408-3333 

2-30 

409 

2-30 

410 

2-31 

412 

2-32 

415 

2-34 

35 

36 

432 

2-37 

433 

2-37 

434 

2-38 

436 

2-39 

439 

2-40 

36 

37 

456-3333 

2-43 

457 

2-44 

458 

2-44 

460 

2-45 

463 

2-47 

37 

38 

481-3333 

2-50 

482 

2-50 

483 

2-51 

485 

2-52 

489 

2-54 

38 

39 

507 

2-57 

508 

2-57 

51  >9 

2-58 

611 

2-59 

515 

2-61 

39 

40 

533-3333 

2-63 

534 

2-64 

535 

2-64 

538 

2-66 

641 

2-67 

40' 

41 

560-3333 

2-70 

561 

2-70 

562 

2-71 

565 

2-72 

569 

2-74 

41 

42 

588 

2-77 

589 

2-77 

590 

2-77 

593 

2-79 

597 

2-81 

42 

43 

616-3333 

2-83 

017 

2-84 

618 

'2-84 

621 

2-86 

625 

2-88 

43 

44 

645-3333 

2-90 

646 

2-90 

648 

2-91 

6.H 

2-92 

655 

2-94 

44 

45 

675 

2-97 

676 

2-97 

677 

2-98 

680 

2-99 

685 

3-01 

45 

46 

7U53333 

3-03 

706 

3-04 

708 

3-04 

711 

3-06 

716 

3-08 

46 

47 

736-3333 

3-10 

737 

3-10 

739 

3-11 

742 

3-13 

747 

3-15 

47 

48 

768 

317 

769 

3-17 

771 

3-18 

774 

3-19 

780 

3-21 

48 

49 

800-3333 

3-23 

801 

3-24 

803 

324 

807 

3-26 

812 

3-28 

49 

50 

833-3333 

330 

834 

331 

836 

331 

840 

333 

846 

335 

50 

Hight 

Hight 

in 
feet. 

Surf.-elope  0°. 

Surf.-elope  5°. 

Surf.-elopelO0. 

Surf.-slope  15°. 

Surf.-slope  20°. 

in 

feet. 

168 


MEASUREMENT    OF  EARTHWORKS. 


Triangular  Areas,  in  square  feet,  for  side-slopes  of  1  to  1,  to  inter- 
section of  slopes,     (r  =  1.)     Slope  angle  =  *45°. 


TABLE  3— Earth-cut. 


Hight 

Surf.-slope  0°. 

Surf.-slope  5°. 

Surf.-slope  10°. 

Surf.-slope  15°. 

Surf.-slope  20°. 

Hight 

in 

in 

feet. 

Pro. 

Pro. 

Pro. 

Pro. 

i'lo. 

feet 

Areas. 

for  -1. 

Areas. 

for  -1. 

Areas. 

for  -1. 

Areas. 

for  -1. 

Areas. 

for  -1. 

1 

1-UOOO 

•10 

1-0077 

.10 

1-0321 

•10 

1-0773 

•11 

1.1527 

•12 

1 

2 

4 

•30 

4 

•30 

4 

•31 

4 

•32 

5 

•35 

2 

3 

9 

•50 

9 

•50 

9 

•52 

10 

•54 

11 

•58 

3 

4 

16 

•70 

16 

•70 

17 

•72 

17 

•75 

18 

•81 

4 

5 

25 

•90 

25 

•90 

26 

•93 

27 

•97 

29 

1-04 

5 

6 

36 

1-10 

36 

1-11 

37 

1-14 

39 

1-19 

42 

1-27 

6 

7 

49 

1-30 

49 

1-31 

51 

1-34 

53 

1-40 

56 

1-50 

7 

8 

64 

1-50 

64 

1-51 

66 

1-55 

69 

1-62 

74 

1-73 

8 

9 

81 

1-70 

82 

1-71 

84 

1-75 

87 

1-83 

93 

1-96 

9 

10 

100 

1-90 

101 

1-91 

103 

1-96 

108 

2-05 

115 

2-19 

10 

11 

121 

2-10 

122 

2-12 

125 

2-17 

130 

2-26 

139 

2-42 

11 

12 

144 

2-30 

145 

232 

149 

2-37 

155 

2-48 

166 

2-65 

12 

13 

169 

2-50 

170 

2-52 

174 

2-58 

182 

269 

195 

2-88 

13 

14 

196 

2-70 

198 

2-72 

202 

2-79 

211 

2-91 

226 

3-11 

14 

15 

225 

2-90 

227 

2-92 

232 

2-99 

242 

3-12 

259 

3-34 

15 

16 

256 

3-10 

258 

3-12 

264 

3-20 

276 

3-34 

295 

357 

16 

17 

289 

3-30 

291 

3-33 

298 

3-41 

311 

3-56 

333 

3-80 

17 

18 

324 

3-50 

327 

3-53 

334 

3-61 

34d 

3-77 

373 

4-03 

18 

19 

361 

3-70 

364 

3-73 

373 

3-82 

389 

3-99 

416 

4-27 

19 

20 

400 

3-90 

403 

3-93 

413 

4-02 

431 

420 

461 

4-50 

20 

21 

441 

4-10 

444 

4-13 

455 

4-23 

475 

4-42 

508 

4-73 

21 

22 

484 

4-30 

488 

4-33 

499 

4-44 

521 

4-63 

558 

4-96 

22 

23 

529 

4-50 

533 

4-53 

546 

4-64 

570 

4-85 

610 

5-19 

23 

24 

576 

4-70 

580 

4-74 

594 

4-85 

621 

5-06 

664 

5-42 

24 

25 

625 

4-90 

630 

4-94 

645 

5-06 

673 

5-28 

720 

5-G5 

25 

26 

676 

5-10 

681 

5-14 

698 

5-26 

728 

5-49 

779 

5-88 

26 

27 

729 

5-30 

735 

5-34 

752 

5-47 

785 

5-71 

840 

6-11 

27 

28 

784 

5-50 

790 

5-54 

809 

5-68 

845 

5-92 

904 

6-34 

28 

29 

841 

5-70 

848 

5-74 

868 

5-88 

906 

6-14 

9C9 

6-57 

29 

30 

900 

5-90 

907 

5-95 

929 

6-09 

970 

6-36 

1037 

6-80 

30 

31 

961 

6-10 

968 

6-15 

992 

6-30 

1035 

6-57 

1108 

7-03 

31 

32 

1024 

6-30 

1032 

6-35 

1057 

6-50 

1103 

6-79 

1180 

7-26 

32 

33 

1089 

6-50 

log- 

6-55 

1124 

6-71 

1173 

7-00 

1255 

7-49 

33 

34 

1156 

6-70 

lies 

6-75 

1193 

6-91 

1245 

722 

1333 

7-72 

34 

35 

1225 

6-90 

1234 

6-95 

1264 

7-12 

1320 

7;43 

1412 

7-95 

35 

36 

1296 

7-10 

1306 

7-15 

1338 

7-33 

1396 

7-65 

1494 

8-18 

36 

37 

1369 

7-30 

1380 

7-36 

1413 

7-53 

1475 

7-86 

1578 

8-41 

37 

38 

1444 

7-50 

1455 

7-56 

1490 

7-74 

1556 

8-08 

1665 

8-64 

38 

39 

1521 

7-70 

1533 

7-76 

1570 

7-95 

11)39 

8-29 

1753 

8-88 

39 

40 

1600 

7-90 

1612 

7-96 

1651. 

8-15 

1724 

8-51 

1844 

9-11 

40 

41 

1681 

8-10 

1694 

8-16 

1735 

8-36 

1811 

8-73 

1938 

934 

41 

42 

1764 

8-30 

1778 

8-36 

1820: 

8-57 

1900 

894 

2033 

9-57 

42 

43 

1S49 

8-50 

T863 

8-56 

1908 

8-77 

1992 

9-10 

2131 

9-80 

43 

44 

19o6    • 

8-70 

1951 

8-77 

1998 

8-98 

2086 

9-37 

2232 

10-03 

44 

45 

2025 

8-90 

2041 

8-97 

2090 

9-18 

2182 

9-59 

2334 

10-26 

45 

46 

2116 

9-10 

2132 

9-17 

2184 

9-39 

2280 

980 

2439 

10-49 

46 

47 

2209 

930 

2226 

9-37 

2280 

9-60 

2380 

10-02 

2546 

10-72 

47 

48 

2304 

9-50 

2322 

9-57 

2378 

980 

2482 

10-23 

2656 

10-95 

48 

49 

2401 

9-70 

2420 

9-77 

2478 

10-01 

2587 

10-45 

2768 

11-18 

49 

50 

2500 

9-90 

2519 

9-97 

2580 

10-22 

•2693 

10-67 

2882 

11-41 

50 

Hight 

liight 

in 
feet. 

Surf.-slope  0°. 

Surf.-slope  5°. 

Surf.-slope  10°. 

Surf.-slope   15°. 

Surf-slope  20°. 

in 
(Vet. 

TABLES. 


169 


Triangular  areas,  in  square  feet,  for  side-slopes  of 
section  of  slopes,     (r  =  H.)     Slope  angle  =  33°  41'. 


to  1,  to  inter- 


TAJ1LE 4— Bank. 


Right 

Surf-slope  0°. 

Surf.-slope  5°. 

Surf.-fllope  10°. 

Surf.-fllope  15°. 

Surf.-slope  30°. 

Hight 

in 

in 

feet. 

Pro. 

Pro. 

Pro. 

Pro. 

Pro. 

feet. 

Areas. 

for  -1. 

Areas. 

for  -1. 

Areas. 

for  -1. 

Areas. 

for  -1. 

Areas. 

for  -1. 

1 

1-5000 

•16 

1-52U7 

•15 

1-6133 

•16 

1-7900 

•18 

2-1378 

•21 

1 

2 

6 

•45 

6 

•46 

6 

•48 

7 

•54 

9 

•6i 

2 

3 

13-5 

•75 

14 

•76 

15 

•81 

16 

•89 

19 

1-07 

3 

4 

24 

1-05 

25 

1-07 

26 

1-13 

29 

1-25 

34 

1-50 

4 

5 

37-5 

1-35 

38 

1-37 

40 

145 

45 

1-61 

54 

1-92 

5 

6 

54 

1-65 

55 

1-68 

58 

1-78 

64 

1-97 

77 

2-35 

6 

7 

73-5 

1-95 

75 

1-98 

79 

2-10 

88 

233 

105 

2-78 

7 

8 

96 

2-25 

98 

2-29 

103 

2-42 

115 

2-68 

137 

321 

8 

9 

121-5 

2-55 

124 

2-59 

131 

2-74 

145 

304 

173 

363 

9 

10 

150 

2-85 

153 

2-90 

161 

3-06 

179 

3-39 

214 

4-06 

10 

11 

181-5 

315 

185 

3-20 

195 

3-39 

217 

3-76 

259 

4-49 

11 

12 

216 

3-45 

220 

351 

232 

3-71 

258 

4-12 

308 

4-92 

12 

13 

253-5 

3-75 

258 

3-82 

273 

403 

302 

447 

361 

5-34 

13 

14 

294 

4-05 

299 

4-12 

316 

4-36 

351 

4-83 

419 

5-77 

14 

15 

337-5 

4-35 

344 

4-43 

363 

4-68 

403 

5-19 

481 

620 

15 

16 

384 

465 

391 

4-73 

413 

5-00 

458 

5-55 

647 

6-63 

16 

17 

433-5 

4-95 

441 

5-04 

466 

-  5-32 

617 

5-92 

618 

7-05 

17 

18 

486 

5-25 

495 

.5-34 

523 

5-65 

580 

626 

693 

7-48 

18 

19 

5415 

5-55 

551 

5-65 

582 

5-97 

646 

6-62 

772 

7-91 

19 

20 

600 

5-85 

611 

5-95 

645 

6-29 

716 

6-98 

855 

8-34 

20 

21 

661-5 

6-15 

673 

6-26 

711 

6-61 

789 

7-34 

943 

8-76 

21 

22 

726 

6-45 

739 

6-56 

781 

6-94 

866 

7-69 

1035 

9-19 

22 

23 

793-5 

6-75 

808 

6-87 

853 

7-26 

947 

8-05 

1131 

962 

23 

24 

864 

7-05 

879 

7-17 

929 

7-58 

1031 

8-41 

1231 

10-05 

24 

25 

937-5 

7-35 

954 

7-48 

1008 

7-90 

1118 

8-77 

1336 

10-47 

25 

26 

1014 

7-6.5 

1032 

7-79 

1090 

8-23 

1210 

9-13 

1445] 

10-90 

26 

27 

1093-5 

7-95 

1113 

8-09 

1176 

8-55 

1304 

9-48 

1558 

1133 

27 

28 

1176 

8-25 

1197 

8-40 

1265 

8-87 

1403 

9-84 

1676 

11-76 

28 

29 

1261-5 

8-55 

1284 

8-70 

1357 

9-19 

1505 

10-20 

1798 

1218 

29 

30 

1350 

8-85 

1374 

»  9-00 

1452 

9-52 

1610 

10-55 

1924 

12-61 

30 

31 

1441-5 

9-15 

1467 

9-31 

1550 

9-84 

1719 

10-91 

2054 

1304 

31 

32 

1536 

9-45 

1563 

9-62 

1652 

10-16 

1832 

11-27 

2189 

13-47 

32 

33 

1633-5 

975 

1662 

9-92 

1757 

10-48 

1948 

11-63 

2328 

13-89 

33 

34 

1734 

10-05 

1765 

10-23 

1865 

10-81 

2068 

1199 

2471 

1432 

34 

35 

l«37-5 

10-35 

1870 

10-53 

1976 

11-13 

2192 

12-35 

2619- 

14-75 

35 

36 

1944 

10-65 

1978 

10-84 

2090 

11-45 

2319 

12-70 

2770 

15-18 

36 

37 

I10&9 

10-95 

2090 

11-14 

2208 

11-77 

2449 

13-06 

2926 

15-60 

37 

38 

2166 

11-23 

2204 

11-45 

2329 

12-10 

2584 

1342 

3087 

1603 

38 

39 

2281-5 

11-55 

2322 

11-76 

2453 

12-42 

2721 

13-78 

3251 

1646 

39 

40 

2400 

11-85 

2442 

12-06 

2581 

12-74 

2863 

14-14 

3420 

16-89 

40 

41 

2521-5 

12-15 

2566 

12-36 

2711 

13-06 

3008 

14-50 

3593 

17-31 

41 

42 

2646 

12-45 

2693 

1267 

2845 

13-39 

3156 

14-85 

3771 

17-74 

42 

43 

2773-5 

12-75 

2823 

1298 

2982 

1371 

3308 

15-21 

3952 

18-17 

43 

44 

2904 

13-05 

2955 

13-28 

3123 

14-03 

3464 

15-57 

4138 

18-60 

44 

45 

3037-5 

13-35 

3091 

13-59 

3266 

14-35 

3623 

15-92 

4329 

1902 

45 

46 

3174 

13-65 

3230 

1389 

3413 

14-68 

3786 

16-28 

4523 

1945 

46 

47 

3313-5 

13-95 

3372 

14-20 

3563 

15-00 

3952 

16-64 

4722 

19-88 

47 

48 

3456 

14-25 

3517 

1450 

3716 

15-32 

4122 

16-99 

4925 

20-31 

48 

49 

3601-5 

;u-55 

3665 

14-81 

3873 

15-64 

4296 

17-35 

5132 

2074 

49 

50 

3750 

14-85 

3816 

15-12 

4032 

15-97 

4473 

17-71 

5344 

2116 

50 

Hight 

Right 

in 

feet. 

Surf.-slope  0°. 

Snrf.-Blope  5°. 

Surf.-slope  10°. 

Surf.-elope  15°. 

Surf.-slope  «0°. 

in 

feet. 

TABLE    OF    CUBIC   YARDS 

IN  FULL  STATIONS,  OR  LENGTHS  OF  WO  FEET. 

CALCULATED  FOR  EVERY  FOOT  AND  TENTH  OF  MEAN  AREA, 
FROM  0-  TO  1000'  SUPERFICIAL  FEET. 


Note. — On  every  page  of  the  Table,  the  columns  on  both  sides  headed  M.A.  contain  the 
Mean  Areas,  in  square,  or  superficial  feet. 

The  horizontal  lines  at  top  and  bottom  show  the  tenths  of  square  feet  of 
Mean  Area. 

And  the  figures  in  the  body  of  the  Table,  computed  to  three  places  of  decimals, 
are  the  Cubic  Yards  (for  100-  feet),  corresponding  to  the  feet  and  tenths  of  Mean 
Area,  indicated  in  the  side  columns,  and  lines  of  tenths  at  top  and  bottom. 


EXPLANATION  OF  THE  TABLE  OF  CUBIC  YARDS, 
To  Mean  Areas,  in  lengtJis  of  100*  feet,  and  of  its  Applications. 

This  Table  is  computed  to  facilitate  the  conversion  into  Cubic  Yards 
of  the  content  of  any  solid  100  feet  in  length,  of  which  the  Mean  Area, 
in  superficial  feet  has  been  ascertained.  It  applies  directly  to  all 
Mean  Areas  from  0'  to  1000'  square  feet  (including  tenths  of  feet), 
and  being  calculated  to  three  decimal  places,  it  extends  indirectly  to 
100,000*  superficial  feet  of  Mean  Area,  as  will  be  shown  hereafter. 


EXAMPLE  1. 

Cubic  yards  for 
full  stations 


To  find  the  Cubic  Yards,  belonging  to  579' 8 
sup.  ft.  of  Mean  Area,  for  a  full  station,  or  length 
of  100-  feet : 

Opposite  579'  and  under  *8  we  find  the  con- 
tent,  or  solidity =2147 '407  cubic  yards. 


Which  is  equal  to 

579- 8  sq.  ft.  of  Mean  Area  X  100'  feet  long, 

and  divided  by  27. 
170 


RULES    FOR   THE    MEASUREMENT   OF    EARTHWORKS. 


171 


EXAMPLE  2. 

Cubic  yards  for 

short  stations 

(-100-) 


EXAMPLE  3. 

Cubic  yards  for 

long  stations 

(+  100-) 


/    Let  the  Mean  Area  of  any  solid,  be  98* T  sq.  ft. 

and  its  length  84  ft.  lineal :  (being  a  short  station). 

Then   at   98' 7  we  find   365*556   cubic   yards, 

which  being  multiplied  by  '84  taken  decimally, 

gives  365-556  X  '84 =307'067  cubic  yards. 

Equal  to...  §*7X§* 


Again,  let  the  Mean  Area  be  88* 6  and  the 
length  259-  feet  (or  a  long  station) ;  then  for  88'  6 
sq.  ft.  of  Mean  Area,  we  have  328*148  cubic 
yards,  which  multiplied  by  2*59  (decimal) 
gives =849-903  cubic  yards. 


Equal  to... 


88-6  X  259- 
27- 


Th  is  Table  is  especially  useful  in  the  computation  of  the  Earth- 
work of  Railroads,  and  other  Public  Works,  where  cross-sections 
have  been  taken  normal  to  a  guide  line,  at  distances  (generally)  of 
100-  lineal  feet  (or  full  stations),  and  the  Mean  Area  calculated  in 
superficial  feet  and  parts:  but  it  is  also  applicable  to  any  solid  of 
which  the  mean  section  is  known  in  square  feet,  and  the  length  100* 
feet,  or  any  decimal  part  thereof. 

For,  if  the  distances  apart  of  cross-sections,  or  lengths  of  stations, 
be  more,  or  less,  than  100'  feet,  we  have  only  to  take  them  decimally, 
as  in  the  above  examples,  and  by  a  simple  multiplication,  of  the 
tabular  quantity,  belonging  to  the  known  area,  the  correct  number 
of  cubic  yards  will  be  ascertained. 

The  Table  being  calculated  to  three  places  of  decimals,  readily  ad- 
mits of  being  used  for  Mean  Areas,  much  exceeding  its  direct  range 
of  1000-  superficial  feet  (as  follows)  : 

EXAMPLE  4.  Suppose  the  Mean  Area  to  be  98,967*  *  sq.  ft.  (repre- 
senting a  cut  98' 9  feet  deep,  and  1000"  feet  wide). 

Then  for  98,900'  (by  moving  the  decimal  point 
of  the  tabular  quantity  of  cubic  yards  for  989' 
two  figures  to  the  right) — 

We  have,  area  98,900'  =  366,296- 3  cubic  yds. 
Add 67-  4=  249- 8  " 


Total,  for  sq.  ft...  98,967'  <=  366,545- • 
Equal  to '- — 


172  RULES  FOR   THE   MEASUREMENT    OF    EARTHWORKS. 

Again,  take  a  Mean  Area,  of  100,048' 9  sq.  ft.  (representing  a  cut 
100-  feet  deep,  and  1000'  feet  wide). 

Then  for  100,000  sq.  ft.  (by  moving  the  deci- 
mal point  of  the  tabular  quantity  of  cubic  yards 
for  1000*  two  figures  to  the  right), 
We  have,  100,000  Area  ==  370,370' 4  cub.  yds. 

Add  48; 9   "     =        181' l    "     " 

Total  for 100,048' 9"     =  370,551' 5    "     " 

Equal  to...  100^X100, 

Example  4,  shows  the  easy  application  of  the  Table,  to  Mean 
Areas,  which  may  be  called  immense,  by  merely  moving  the  decimal 
point,  and  a  simple  addition,  as  shown  above. 

Other  methods  of  using  the  Table  will  occur  to  the  reader,  but  the 
examples  given  seem  sufficient  for  illustration. 

Much  pains  have  been  taken  to  make  this  Table  correct,  to  the 
nearest  decimal,  and  we  believe  it  may  be  safely  depended  on. 

Note. — Besides  its  special  application  to  Earthworks,  the  extensive 
Table  following  is  also  a  general  Table  for  the  conversion  of  any  sum 
of  Cub.ic  Feet  into  Cubic  Yards.  Thus,  in  the  example  at  page  103, 
the  reduced  quantities  of  Cubic  Feet  sum  up  227,200  —  30,000  = 
197,200  Cubic  Feet. 

In  such  cases  we  have  only  to  cut  off  two  figures  from  the  right 
(or  H-  by  100),  and  we  have  1972,  the  mean  area,  which,  in  100  feet 
length,  would  have  produced  the  quantity  given. 

With  197*2  we  enter  the  Table  following,  and  find  730'370  Cubic 
Yards ;  now,  moving  the  decimal  point  one  place  to  the  right,  we 
have  7303-70  Cubic  Yards,  or  in  round  numbers,  7304  Cubic  Yards, 
as  already  given  on  page  103. 

In  like  manner  the  Cubic  Yards  for  any  sum  whatever  of  Cubic 
Feet  can  readily  be  obtained,  and  the  Table  being  in  itself  strictly 
correct,  the  result  will  be  reliable. 


RULES   FOR    THE    MEASUREMENT    OF    EARTHWORKS. 


173 


TABT.E   OF  CUIilC  YARDS,  in  full  Station*,  or  length  ft  of  1OO  fret:  for  every 
foot  ami  ti-nth  of  Menu  Arm,  from  O  to  1OOO  Snjurftcinl  Fret. 


M.A. 

•0 

•  1 

•a 

•3 

•4 

•5      |      -6 

•7 

•8 

•9 

M.A. 

0 

o-ooo 

0-370 

0-741 

1-111 

1-481 

1-852 

2-222 

2-593 

2-963 

3-333 

0 

1 

3-704 

4-074 

4-444 

4-815 

5-185 

5-556 

5-926 

6-296 

6-667 

7-037 

1 

2 

7-407 

7-778 

8-148 

8-519 

8-889 

9-259 

9630 

10- 

10.370 

10741 

2 

3 

11-111 

11481 

11-852 

12-222 

12-593 

12-963 

13-333 

13-704 

14-074 

14-444 

3 

4 

14-815 

15-185 

15-556 

15-920 

16-296 

16-667 

17-037 

17-407 

17-778 

18148 

4 

5 

18-519 

18-889 

19-259 

19-630 

20- 

20370 

20-741 

21-111 

21481 

21-852 

5 

6 

22-222 

22-593 

22-963 

23-333 

23-704 

24-074 

24-444 

24-815 

25-185 

25-556 

6 

7 

25920 

26-296 

26-667 

27037 

27-407 

27-778 

28-148 

28519 

28-889 

29-259 

7 

8 

29-630 

30- 

30-370 

30-741 

31-111 

31-481 

31-852 

32-222 

32-593 

32963 

8 

9 

33-333 

33-704 

34-074 

34-444 

34-815 

35-185 

35-556 

35-92b 

36-296 

36-667 

9 

10 

37037 

37-407 

37-778 

38-148 

38-519 

38-889 

39-259 

39-630 

40- 

40-370 

10 

11 

40-741 

41-111 

41-481 

41-852 

42-222 

42-593 

42-963 

43-333 

43704 

44-074 

11 

12 

44444 

44-815 

45-185 

45-556 

45-926 

46-296 

46-667 

47-037 

47-407 

47-778 

12 

13 

48-148 

48-519 

48-889 

49-259 

49630 

50 

60-370 

60-741 

51-111 

51-481 

13 

14 

51-852 

52  222 

52-593 

52-963 

53333 

53-704 

64-074 

54-444 

54*815 

55-185 

14 

15 

55-556 

55-926 

56-296 

56-667 

57-037 

57-407 

57-778 

68-148 

58-519 

68-889 

15 

16 

59-259 

59-630 

60- 

60-370 

60-741 

61-111 

61-481 

61852 

02-222 

62-593 

16 

17 

62-903 

63-3:53 

63-704 

64-074 

64-444 

64815 

65185 

65-556 

65-926 

66-296 

17 

18 

66-667 

67.037 

67-407 

67-778 

68-148 

(8-519 

68-889 

69-259 

69-630 

70- 

18 

19 

70-370 

70-741 

71-111 

71-481 

71-852 

72-222 

72-593 

72-963 

73-333 

73-704 

19 

20 

74-074 

74-444 

74-815 

75-185 

75-556 

75-926 

76-29G 

76-667 

77-037 

77-407 

20 

21 

77-778 

78-148 

78-519 

78-889 

79-259 

79-630 

80- 

80-370 

80-741 

81111 

21 

22 

81-481 

81-852 

82-222 

82593 

82-963 

83-3:  J3 

83-704 

84-074 

84-444 

84-815 

22 

23 

85-185 

85-556 

85-92f. 

86-296 

86-667 

87-037 

87-407 

87-778 

88-148 

88-519 

23 

24 

88-889 

89-259 

89-630 

9J- 

90-370 

90-741 

91-111 

91-481 

91-852 

92-222 

24 

25 

92-593 

92-963 

93-333 

93.704 

94-074 

94-444 

94-815 

95-185 

96-556 

95926 

25 

26 

96-296 

96-667 

97-037 

97-407 

97-778 

98-148 

985i  9 

98-889 

99-259 

99-630 

26 

27 

100- 

100.370 

100741 

101-111 

101-481 

101-852 

102-222 

102.593 

102-963 

103-333 

27 

28 

103-704 

104-074 

104-444 

104-815 

105-186 

105-556 

105-920 

106-296 

106-667 

107-037 

28 

29 

107-407 

107-778 

108-148 

108-519 

108-889 

109-259 

109-630 

110- 

110-370 

110-741 

29 

30 

Ill-Ill 

111-481 

111-852 

112-222 

112-593 

112-963 

113-333 

113-704 

114-074 

114-444 

30 

31 

114-815 

115-185 

115-556 

115-926 

116-296 

116-667 

117-037 

117-407 

117-778 

118-148 

31 

32 

118-519 

118-889 

119259 

119-630 

120- 

120-370 

120-741 

121-111 

121-481 

121-852 

32 

33 

122-222 

122-593 

122-963 

123-333 

123-704 

124-074 

124-444 

124-815 

125-lSo 

125*-556 

33 

34 

125-92b 

126-296 

126-667 

127-037 

127-407 

127-778 

128-148 

128-519 

128-889 

129-259 

34 

35 

129-630 

130- 

130-370 

130-741 

131-111 

131-481 

131-852 

132-222 

132-593 

132.963 

35 

36 

133-333 

133-704 

134-074 

134-444 

134-815 

135-185 

135-556 

135-926 

136-296 

136-667 

36 

37 

137-037 

137-40" 

137-778 

138-148 

138-519 

138-889 

139-259 

139-630 

140- 

140-370 

37 

38 

140-741 

141-111 

141-481 

141-852 

142-222 

142-593 

142-963 

143-33o 

143-704 

144-074 

38 

39 

144-444 

144  815 

145-185 

145-556 

145-926 

146-296 

146-667 

147-03" 

147-407 

147-778 

39 

40 

148-148 

148-519 

148-889 

149-259 

149-630 

150- 

150-37 

160-74 

161-11 

151-481 

40 

41 

151-8&* 

152-22 

152-593 

152-963 

153-333 

153-704 

154-07 

154-444 

154-81 

155-185 

41 

42 

155-55 

15592 

156296 

156-66' 

157-037 

157-407 

157-77 

158-14 

168-51 

168-889 

42 

4;i 

159-25 

159-63 

160- 

160-370 

160-741 

161-11 

161-48 

161-85 

162-222 

162-593 

43 

44 

162-%. 

163-333 

163-704 

164-074 

164-444 

164-81 

165-18 

165-55 

165-92 

166-296 

44 

45 

16666 

107-03 

167-40" 

167-77S 

168-148 

168-61 

168-88 

169-25 

169-630 

170- 

46 

46 

170-37 

170-74 

171-111 

171-481 

171-852 

17222 

172-59 

172-96. 

173-33 

173-704 

46 

47 

174-07 

174-444 

174-815 

175-18o 

175-556 

175-92 

176-29 

176-66 

177-03 

177-40 

47 

43 

177-77 

178-14 

178-519 

178-889 

179-259 

179-630 

180- 

18037 

180-74 

181-111 

48 

49 

181-48 

181-85 

182-222 

182-593 

182-963 

183-333 

183-70 

184-07 

184-444 

184-815 

49 

50 

185-lSa 

185-55 

185-926 

186-29b 

186-667 

187-03 

187-40 

187-77 

188-14 

188-519 

60 

i 

51 

188-88 

189-259 

189-630 

190- 

190-37 

190-74 

191-11 

191-48 

191-85 

192-222 

51 

52 

192-59 

192-96 

193-333 

193-704 

194-074 

194-444 

194-81 

195-18 

195-55 

195-926 

62 

196-29 

196-66 

197-037 

197-407 

197-77 

198-14 

198-51 

198-88 

199-25 

199-630 

53 

54 

200- 

200-37 

200-74 

201-11 

201-48 

201-85 

202-  22* 

202-59 

202-96 

203-333 

54 

55 

203-70 

204-07 

204-444 

204-81 

205-18 

205-55 

205-92 

206-29 

206-66 

207  03" 

65 

56 

207-40 

207-77 

208-14 

208-51 

208-88 

209-25 

209-630 

210- 

210-37 

210-74 

56 

57 

211-11 

211-48 

211-85 

212-222 

212-59 

212-96, 

213-333i   213-704 

214-07 

214-444 

67 

58 

214-81 

215-18 

215-55 

215-92 

216-29 

216-66 

217-037 

217-40 

217-77 

218-14 

58 

59 

218-51 

218-88 

219-25 

219-63 

220- 

220-37 

220-741 

221-11 

221-48 

221-85 

59 

CO 

222-22 

222-59 

222-96- 

223-333 

223-70 

224-07 

224-444 

224-81 

225-18 

225-55 

60 

M.A 

•0 

•1 

•a 

•3 

•4 

•5 

•6 

•7 

•8 

•9 

M.A. 

MEAN  AREAS   O  to  6O. 

174 


RULES   FOR   THE   MEASUREMENT    OF   EARTHWORKS. 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  WO  FEET  IN  LENGTH. 


M.A. 

•0 

•1 

•3 

•3 

•4: 

•5 

•6 

•7 

•8 

•9 

M.A. 

61 

225-926 

226-296 

226-667 

227-037 

227-407 

227-778 

228-148 

228-519 

228-889 

229-259 

61 

62 

229-630 

230- 

230-370 

230-741 

231-111 

231-481 

231-852 

232-222 

232-593 

232-963 

62 

63 

233-333 

233-704 

234-074 

234-444 

234-815 

235-185 

235-556 

235-926 

236-296 

236-667 

63 

64 

237-037 

237-407 

237-778 

238-148 

238-519 

238-889 

239-259 

239-630 

240- 

240-370 

64 

65 

240741 

241-111 

241-481 

241-852 

242-222 

242-593 

242-963 

243-333 

243704 

244-074 

65 

66 

244-444 

244-815 

245-185 

245-556 

245-926 

246-296 

246-667 

247-037 

247-407 

247-778 

66 

67 

248-148 

248519 

248-889 

249-259 

249-630 

250- 

250-370 

250-741 

251-111 

251-481 

67 

6S 

251-852 

252-222 

252-593 

252-963 

253-33o 

253-704 

254-074 

254-444 

254-815 

255-185 

68 

69 

255-556 

255-926 

256-296 

256-667 

257-037 

257-407 

257-778 

258-148 

258-519 

25K-889 

C9 

70 

259-259 

259-630 

260- 

260-370 

260-741 

261-111 

261-481 

261-852 

262-222 

262-593 

70 

71 

262-963 

263-333 

263-704 

264-074 

264-444 

264-815 

265-185 

265-556 

265-926 

266-296 

71 

72 

266-667 

267-037 

267-407 

267-778 

268-148 

268-519 

268-889 

269-259 

269-630 

270- 

72 

73 

270-370 

270741 

271-111 

271-481 

271-852 

272-222 

272-593 

272-963 

273-333 

273-704 

73 

74 

274-074 

274-444 

274-815 

275-185 

275-556 

275-926 

276-296 

276-667 

277-037 

277-407 

74 

75 

277-778 

278-148 

278-519 

278-889 

279-259 

279-630 

280- 

280-370 

280-741 

281-111 

75 

76 

281-4S1 

281-852 

282-222 

282-593 

282-963 

283  333 

283-704 

284-074 

284-444 

284-815 

76 

77 

•285-185 

285-556 

285-926 

286-296 

286-667 

287-037 

287-407 

287-778 

288-148 

288-519 

77 

78 

288-889 

289-259 

289-630 

290- 

290-370 

290-741 

291-111 

291-481 

291-852 

292-222 

78 

79 

292-593 

292-963 

293-333 

293-704 

294-074 

294-444 

294-815 

'295-185 

295-556 

295-926 

79 

80 

296-296 

296-667 

297-037 

297-407 

297-778 

298-148 

298-519 

298-889 

299-259 

299-630 

80 

81 

300- 

300-370 

300-741 

301-111 

301-481 

301-852 

302-222 

302-593 

302-963 

303-333 

81 

82 

303-704 

304-074 

304444 

304-815 

305-185 

305-556 

305-926 

306-296 

306-667 

307  037 

82 

83 

307-407 

307-778 

308-148 

308-519 

308-889 

309-259 

309-630 

310- 

310-370 

310-741 

83 

84 

311-111 

311-481 

311-852 

312-222 

312-593 

312-963 

313-333 

313-704 

314-074 

314-444 

84 

85 

314-815 

315-185 

315-556 

315-926 

316-296 

316-667 

317-037 

317-407 

317-778 

318-148 

85 

86 

318-519 

318-889 

319-259 

319-630 

320- 

320-370 

320-741 

321-111 

321-481 

321-852 

86 

87 

322-222 

322-593 

322-963 

323-333 

323-704 

324-074 

324-444 

324-815 

325-185 

325-556 

87 

88 

325'  926 

320-296 

326-667 

327-037 

327-407 

327-778 

328-148 

328-519 

328-889 

329-259 

88 

89 

329-630 

330: 

330-370 

330-741 

331-111 

331-481 

331-852 

332-222 

332-593 

332-963 

89 

90 

333-333 

333-704 

334-074 

334-444 

334-815 

335-185 

335-556 

335-926 

336-296 

336-667 

90 

91 

337-037 

337-407 

337-778 

338-148 

338-519 

338-889 

339-259 

339-630 

340- 

340-370 

91 

92 

340-741 

341-111 

341-481 

341-852 

342-222 

342-593 

342-963 

343-333 

343-704 

344-074 

92 

93 

344-444 

344-815 

345-185 

345556 

345-926 

346-296 

346-667 

347-037 

347-407 

347-778 

93 

94 

348-148 

348-519 

348-889 

349-259 

349-630 

350- 

350'370 

350-741 

351-111 

351-481 

94 

95 

351-852 

352-222 

352-593 

352-963 

353-333 

353-704 

354-074 

354-444 

354-815 

355-185 

95 

96 

355-556 

355-926 

356-296 

356-667 

357-037 

357-407 

357-778 

358-148 

358-519 

358-889 

96 

97 

359-259 

359-630 

360- 

360-37C 

360-741 

361-111 

361-481 

361-852 

362-222 

362-593 

97 

98 

362-963 

363-333 

363-704 

364-074 

364-444 

364-815 

365-185 

365-556 

365-926 

366-296 

98 

99 

366-667 

367-037 

367-407 

367-778 

368-148 

368-519 

368-889 

369-259 

369-630 

370- 

99 

100 

370-370 

370-741 

371.111 

371-481 

371-852 

372-222 

372-593 

372-963 

373-333 

373-704 

100 

101 

374-074 

374-444 

374-815 

375-185 

375-556 

375-926 

376-296 

376-667 

377-037 

377-407 

101 

102 

377-778 

378-148 

378-519 

378-889 

379-259 

379-630 

380- 

380-370 

380-741 

381-111 

102 

103 

381-481 

381-852 

382-222 

382593 

382-963 

383-333 

383-704 

384-074 

384-444 

384-815 

103 

104 

385185 

385-55b 

385-926 

386-296 

386-667 

387-037 

387-407 

387-778 

388-148 

388-519 

104 

105 

388-889 

389-259 

389-630 

390- 

390-370 

390-741 

391-111 

391-481 

391-852 

392-222 

105 

106 

392-593 

392-963 

393-333 

393-704 

394-074 

394-444 

394-815 

395-185 

395-556 

395-926 

106 

107 

390-296 

396-667 

397-037 

397-407 

397-778 

398-148 

398-519 

398-889 

399-259 

399630 

107 

108 

400- 

400-370 

400-741 

401-111 

401-481 

401-852 

402-222 

402-593 

402-963 

403-333|  108 

109 

403-704 

404-074 

404-444 

404-815 

405-185 

405-556 

405-926 

406-296 

406-667 

407-037 

109 

110 

407-407 

407-778 

408-148 

408-519 

408-889 

409-259 

409-630 

410- 

410-370 

410-741 

110 

111 

411-111 

411-481 

411-852 

412-222 

412-593 

412963 

413-333 

413-704 

414-074 

414-444 

111 

112 

414-815 

415-185 

415-55C 

415-926 

416-296 

416-667 

417-037 

417-407 

417-778 

418-148 

112 

113 

418-519 

418-889 

419-259 

419-600 

420- 

420-370 

420-741 

421-111 

421-481 

421-852 

113 

114 

422-222 

422-593 

422-963 

423-333 

423-704 

424-074 

424-444 

424-815 

425-185 

425-556 

114 

115 

425-926 

426-296 

426-667 

427-037 

427-40- 

427-778 

428-148 

428-519 

428-889 

429-259 

115 

116 

429-630 

430- 

430-370 

430-741 

431-11 

431-481 

431-852 

432-222 

432-593 

432-963 

116 

117 

433-333 

433-704 

434-074 

434-444 

434-815 

435-185 

435-556 

435-926 

436-296 

436-667 

117 

118 

437-03" 

437-407 

437-778 

458-148 

438-519 

438-889 

439-259 

439-630 

440- 

440-370 

118 

119 

440-741 

441-111 

441-481 

441-852 

442-222 

442-593 

442-963 

443-333 

443-704 

444-074 

119 

120 

444-444 

444-815 

445-185 

445-556 

445-926 

446-296 

446-667 

447-037 

447-407 

447-778 

120 

M.A 

•0 

•1 

•a 

•3 

•4 

•5 

6 

•7 

•8 

•9 

M.A. 

•                   MEAN  AREAS  61  to  12O. 

RULES   FOR   THE   MEASUREMENT    OF   EARTHWORKS. 


175 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  1OO  FEET  Iff  LENGTH. 


M.A. 

•0 

•1 

•a 

•3 

•4: 

•5 

•6 

•7 

•8 

•9 

M.A. 

121 

448-148 

448-519 

448-889 

449259 

449.630 

450- 

450-370 

450-741 

451-111 

451-481 

121 

122 

451-852 

452-222 

452-593 

452-963 

453-333 

453-704 

454-074 

454-444 

454-815 

455-185 

122 

123 

455-556 

455-926 

456-296 

456-667 

457-037 

457-407 

457-778 

458-148 

458-519 

458-889 

123 

124 

459-259 

459-630 

460- 

400-370 

460-741 

461-111 

461-481 

461-852 

462-222 

462593 

124 

125 

462-963 

463-333 

463-704 

464-074 

464-444 

464-815 

465-185 

465-556 

465-926 

466-296 

125 

126 

466-667 

467-037 

467-407 

4H7-778 

468-148 

468-519 

468-889 

469-259 

469-630 

470- 

12(1 

127 

470  370 

470-741 

471-111 

471-481 

471-852 

472--222 

472-593 

472-963 

473-333 

473-704 

127 

128 

474-074 

474-444 

474-815 

475-185 

475-556 

475-926 

476-296 

476-667 

477-037 

477-407 

128 

129 

477-778 

478-148 

478-519 

478-889 

479-259 

479-630 

480- 

480-370 

480-741 

481-111 

129 

130 

481-481 

481-852 

482-222 

482593 

482-963 

483-333 

483-704 

484-074 

484-444 

484-815 

130 

131 

485-185 

485556 

485-926 

486-296 

486-667 

487-037 

487-407 

487-778 

488-148 

488-51  9 

131 

132 

488-889  1  489-259 

489-630 

490- 

490-370 

490-741 

491-111 

491-481 

491-852 

492-222 

132 

133 

492-593 

492-963 

493-333 

493-704 

494-074 

494-444 

494-815 

495-185 

495-656 

495926 

133 

134 

496-296 

495-667 

497-037 

497-407 

497-778 

498-148 

498-519 

498-8S9 

499-269 

499-630 

134 

135 

500- 

500-370 

500-741 

501-111 

501-481 

601-852 

502222 

602-593 

602-9B3 

503-333 

135 

136 

503-704 

504-074 

504444 

504-815 

605-185 

605-556 

505926 

506-296 

506-667 

507-037 

136 

137 

5D7-407 

507-778 

508-148 

608-519 

508-889 

609-259 

609630 

610- 

610-370 

510-741 

137 

138 

511-111 

511-481 

511-852 

512-J-22 

512-593 

512-963 

613-333 

513-704 

514-074 

514-444 

138 

139 

514-815 

515-185 

615-556 

515-926 

516-296 

516-667 

517-037 

617-407 

517778 

518-148 

139 

140 

518-519 

518-889 

519-259 

519-630 

520- 

520-370 

520-741 

521-111 

521-481 

521-852 

140 

Ul 

522-222 

522-593 

522-963 

523-333 

523-704 

524-074 

624-444 

524-815 

625-185 

625-556 

141 

142 

525-926 

526-296 

526-667 

527-037 

527-407 

627-778 

628-148 

628-519 

528-889 

529-259 

142 

143 

529-630 

530- 

530-370 

630-741 

531111 

631-481 

631-852 

632-222 

632-593 

632-9C3 

143 

144 

633-333 

533-704 

534-074 

534-444 

634-815 

635-185 

635-556 

635926 

636296 

536-667 

144 

145 

537-037 

537-407 

537-778 

538148 

538-519 

538-889 

539.259 

639-030 

640- 

540-370 

145 

146 

540-741 

541-111 

541-481 

541-852 

542-222 

642-593 

642-963 

543333 

643-704 

544-074 

146 

147 

544-444 

544-815 

545-185 

545-556 

545-926 

646-296 

546-667 

547-037 

547-407 

547-77S 

147 

148 

548-148 

548-519 

648-889 

549-259 

549-630 

650- 

550-370 

650-741 

651-111 

551-481 

148 

149 

551-852 

552-222 

552-593 

652-963 

553-333 

553-704 

554-074 

654-444 

554-815 

655-185 

149 

150 

555-556 

555-926 

656-296 

556-667 

557-037 

657-407 

657-778 

658-148 

658-519 

658-889 

150 

151 

559-259 

559-630 

560- 

560-370 

560-741 

661-111 

661-481 

561-862 

662-222 

662-693 

151 

152 

562-963 

563-333 

663-704 

564-074 

664-444 

664-815 

665-185 

565556 

665-926 

566-296 

152 

153 

566-667 

567-037 

567-407 

567-778 

568-148 

668-519 

568-889 

569-259 

669-630 

670- 

153 

154 

570-370 

570-741 

571-111 

571-481 

671-852 

672-222 

672-593 

672-963 

573-333 

573704 

154 

155 

574-074 

574-444 

574-815 

575-185 

675-556 

675-926 

676-296 

676-607 

677  037 

577-407 

165 

156 

577-778 

578-148 

578-519 

678-889 

579-259 

679-630 

MO 

580-370 

580-741 

581-111 

156 

157 

681-481 

581-852 

582-222 

582-593 

582-963 

583333 

683-704 

684-074 

684-444 

684-815 

157 

158 

585-185 

585-556 

585-926 

586-296 

586-667 

5S7-037 

587-407 

587-778 

588-148 

588-519 

158 

159 

588-889 

589-259 

589-630 

590- 

590-370 

690-741 

691-111 

691-481 

691-852 

592-222 

159 

160 

592-593 

592-963 

593*333 

593-704 

594  074 

594-444 

594815 

595-185 

595-556 

595-926 

160 

161 

596-296 

596-667 

597-037 

597-407 

597-778 

598-148 

698-519 

698-889 

599-259 

699630 

161 

162 

600- 

600-370 

600-741 

601-111 

601-4S1 

601-852 

602-22'J 

602-593 

6U2-963 

603-333 

162 

163 

603-704 

604-074 

604-444 

604-815 

605-185 

605-556 

C05-926 

606-296 

606-667 

607-037 

163 

164 

607-407 

607-778 

608-148 

608-519 

608-889 

609-259 

609-630 

610- 

610-370 

610-741 

164 

165 

611-111 

611-481 

611-852 

612-222 

612-593 

612-963 

613-3*33 

613-704 

614-074 

614-444 

165 

166 

614-815 

615-185 

615556 

615-926 

616-296 

616-667 

617-037 

617-407 

617-778 

618-148 

166 

167 

618519 

618-889 

619  259 

619-630 

620- 

620-370 

620-741 

621-111 

621-481 

621-852 

167 

168 

622  222 

622-593 

622-963 

623-333 

62:1-704 

624-074 

624444 

624-815 

625-185 

625556 

168 

169 

625-926 

626-296 

626-667 

627  -037 

627-407 

627-778 

628-148 

628-519 

628-889 

629-259 

169 

170 

629-630 

630- 

630-370 

630741 

631-111 

631-481 

631-852 

632-222 

632-593 

632-963 

170 

171 

633-333 

633-704 

634-074 

634-444 

634-815 

635-185 

635-556 

635-926 

636-296 

636-667 

171 

172 

637-037 

637-407 

637-778 

638-148 

638-519 

638-889 

639-259 

639-630 

640- 

640-370 

172 

173 

640-741 

641-111 

641-481 

641-852 

642-222 

642-593 

642-963 

643  333 

643-704 

644-074 

173 

174 

644-444 

644-815 

645-185 

645-556 

645-926 

646-296 

646-667 

647-037 

647-407 

647-778 

174 

175 

648-148 

648519 

648-889 

649-259 

649-630 

650- 

650-370 

650-741 

651-111 

651-481 

175 

176 

651-852 

652-222 

652-593 

652963 

653-333 

653-704 

654-074 

654-444 

654-815 

655-185 

176 

177 

655-556 

655-926 

656-296 

656-667 

657-037 

657-407 

657-778 

658-148 

658-519 

658-889 

177 

178 

659-259 

659-630 

660- 

660-370 

660-741 

661-111 

661-481 

661-852 

662-222 

662-593 

178 

179 

662-963 

663-S33 

663-704 

664-074 

664-444 

664-815 

665-185 

665-556 

665-926 

666-296 

179 

180 

666-667 

667-037 

667-407 

667-778 

668-148 

668  519 

668-889 

669-259 

669-630 

670- 

180 

M.A 

•0 

•1 

•a 

•3 

•4 

•5 

•6 

•7 

•8 

•9 

M.A. 

MEAN  AREAS  121  to  ISO. 

176 


RULES    FOR    THE    MEASUREMENT    OF    EARTHWORKS. 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  WO  FEET  IN  LENGTH. 


M.A. 

•O 

•1 

til 

•a 

•4: 

•5 

•6 

•7 

•8 

•9 

M.A. 

181 

670-370 

670-741 

671-111 

671-481 

€71-852 

672-222 

672-593 

672-963 

673-333 

673-704 

181 

182 

674-074 

674-444 

674-815 

675-185 

675-556 

675-926 

676-296 

676-667 

677-037 

677-407 

182 

183 

677-778 

678-148 

678-519 

678889 

679259 

679-630 

680- 

680-370 

680-741 

681-111 

183 

184 

681-481 

681-852 

682-222 

68-2-593 

682-963 

6-3-333 

683-704 

684-074 

684-444 

684-815 

184 

185 

685-185 

685-556 

685-926 

686-296 

686-667 

687-037 

687-407 

687-778 

688-148 

688-519 

185 

186 

688-889 

689-259 

689  630 

690- 

690-370 

690-741 

691-111 

£91-481 

691-852 

692-222 

186 

187 

692-593 

692-963 

693-333 

693704 

694  074 

694444 

694-815 

695-185 

695-556 

695-926 

187 

188 

696-296 

696-667 

697-037 

697-407 

697-778 

698-148 

69S-519 

698-889 

699-259 

699-630 

188 

189 

700- 

700-370 

700-741 

701-111 

701-481 

701-852 

702-22-2 

702-593 

702-963 

703-333 

189 

190 

703-704 

704-074 

704-444 

704-815 

705-185 

705-556 

705-926 

706-296 

706-667 

707-037 

190 

191 

707-407 

707-778 

708-148 

708-519 

708-889 

709-259 

709-630 

710- 

710370 

710-741 

191 

192 

711-111 

711-481 

711-852 

712-222 

712-593 

712-96:^ 

713-333 

713-704 

714-074 

714-444 

192 

193 

714-815 

715-185 

715-556 

715-926 

716-296 

716-667 

717-037 

717-407 

717-778 

718-148 

193 

194 

718-519    718-889 

719;259 

719-630 

720- 

720-370 

7'20-741 

721-111 

7'21-481 

721-852 

194 

195 

722-222 

722-593 

722-963 

723-333 

723-704 

724-074 

724-444 

724-815 

725-185 

725-55fi 

195 

196 

725-926 

726-296 

726-667 

727037 

727-407 

727-778 

728-148 

728-519 

728-F89 

729-259 

196 

197 

729-630 

730- 

730-370 

730-741 

731-111 

731-481 

731-852 

732-222 

732-593 

732963 

197 

198 

733-333 

733-704 

734-074 

734-444 

734815 

735-185 

735-556 

735-926 

736-296 

736-667 

198 

199 

737-037 

737-407 

737-778 

738-148 

738-519 

738-889 

739-259 

739-630 

740- 

740-370 

199 

200 

740-741 

741-111 

741-481 

741-852 

742-222 

742-593 

742-963 

743333 

743-704 

744-074 

200 

201 

744-444 

744-815 

745-185 

745-556 

745-926 

746-296 

746667 

747-037 

747-407 

747-778 

201 

202 

748-148 

748-519 

748-889 

749-259 

749-631 

750- 

750-370 

750-741 

751-111 

751-481 

202 

203 

751-852 

752-222 

752-593 

752-963 

753-333 

753-704 

754-074 

754-444 

754-815 

755-185 

203 

204 

755-556 

755-926 

756-296 

756-667 

757-037 

757-407 

757-778 

758-148 

758-519 

758  889 

204 

205 

759-259 

759-630 

760 

760-370 

760-741 

761-111 

761-481 

761-852 

762-222 

76-2-593 

205 

206 

762-963 

763-333 

763-704 

764-074 

764-444 

764-815 

765-185 

765556 

765-92P 

766-296 

206 

207 

766-667 

767-037 

767-407 

767-778 

768-148 

768-519 

768-889 

769-259 

769-63( 

770- 

207 

208 

770-370 

770741 

771-111 

771-481 

771-85-2 

772-222 

772-593 

772-963 

773-333 

773-704 

208 

209 

774  074 

774-444 

774-815 

775-185 

775-556 

775-920 

776-296 

776-667 

777-037 

777-407 

209 

210 

777-778 

778-148 

778-519 

778-889 

779-259 

779-630 

780- 

780-370 

780-741 

781-111 

•210 

211 

781-481 

781-852 

782-222 

782-593 

782-963 

783-333 

783-704 

784-074 

784-444 

784-815 

'211 

212 

785-185 

785-556 

785-926 

786-296 

786-667 

787-03' 

787-40' 

787-778 

788-148 

788-519 

212 

213 

788-889 

789-259 

789-630 

790- 

790-370 

790-74 

791-11 

791-481 

791-85^ 

792-22'- 

213 

214 

792-593 

792-963 

793333 

793-704 

794-074 

794-444 

794-81o 

795-185 

795-556 

795-9-20 

214 

215 

796-296 

796667 

797-037 

797-407 

797-778 

798-14 

79S-519 

798-889 

799  259 

799-630 

215 

216 

800- 

800-37( 

800-741 

801-111 

801-481 

801-85 

802-22 

802-593 

802-963 

8U3-333 

216 

217 

803-704 

804-074 

804-444 

804-815 

805-185 

80555 

805-92 

806-296 

80606" 

807-037 

217 

218 

807-407 

807-778 

808-148 

808-519 

808-889 

809-25 

809-63 

810- 

810-370 

810-741 

218 

219 

811-111 

811-481 

811-852 

812-222 

812-593 

812-96 

813-33 

813-704 

814-074 

814-444 

219 

220 

814-815 

815-185 

815-556 

815-926 

816-296 

816-66 

817-03 

817-407 

817-77 

818-148 

220 

221 

818-519 

818-889 

819-259 

819-630 

820- 

820-37 

820-74 

821-111 

821-48 

821-85'' 

221 

222 

822-22- 

822-593 

822-963 

823-333 

823-704 

824-07 

824-44 

824-815 

825-18 

825-55 

222 

223 

825-926 

826-29 

826-667 

827-037 

827-407 

827-77 

828-14 

828-519 

828-88 

829-259 

223 

224 

829-630 

830- 

830-370 

830-741 

831-111 

831-48 

831-85 

832-222 

832-59 

832-96 

224 

225 

833-333 

833-70 

834-074 

834-444 

834-815 

835-18 

835-55 

835-926 

836-29 

836-66 

225 

226 

837-037 

837-40 

837-778 

838-148 

838-519 

838-88 

839-25 

839-630 

840- 

840  37( 

226 

227 

840-741 

841-11 

841-481 

841-852 

842-222 

842-59 

842-963 

843-H33 

843-70 

844-07 

227 

228 

844-444 

844-81 

845-185 

845-556 

845926 

846-29 

846-66 

847-037 

847  -JO 

847-77 

228 

229 

848-148 

848-51 

848-889 

849-259 

849-630 

850- 

850-37 

850-741 

851-11 

851-48 

229 

230 

851-852 

852-22 

852-593 

852-963 

853-333 

85370 

854-07 

854-444 

854-81 

855-18 

230 

231 

855-556 

855-92 

856-296 

856-667 

857-037 

857-40 

857-77 

858-148 

858-51 

858-889 

231 

232 

859259 

859-63 

860- 

860-370 

860-741 

861-11 

861-48 

861-852 

862-22 

862-593 

232 

233 

862-963 

86333 

863-704 

864-074 

864-444 

864-81 

865  18. 

865-55G 

865-92 

866-296 

233 

234 

866-667 

867-03 

867-407 

867-778 

868-148 

868-51 

868-8« 

869-259 

869-63 

870- 

234 

235 

870-370 

870-74 

871-111 

871-481 

871-85 

872-22 

872-59 

872-963 

873-33 

873-704 

235 

2:56 

874-074 

874-44 

874-815 

875-185 

875-556 

875-92 

876-29 

876-667 

877-03 

877-407 

236 

237 

877-778 

878-14 

878-519 

878-889 

879-250 

879-63 

880- 

880-370 

880-74 

881-111 

237 

238 

881-481 

881-85 

882-222 

882-593 

882-96: 

883-33 

883-70 

SS4-074 

884-444 

884-815 

238 

239 

885-185 

885-55 

885-926 

886-29e 

886-667 

887-03 

887-40 

887-778 

888-14 

888-51S 

239 

240 

888-889 

889-25 

8S9-63C 

890- 

890-37C 

890-74 

891-11 

891-481 

891-85 

892-222 

240 

M.A 

•O 

•1 

•ft 

•3 

•4 

•5 

•6 

•7 

•8 

•9 

M.A. 

MEAN  AREAS   181  to  24O. 

RULES   FOR   THE    MEASUREMENT    OF   EARTHWORKS. 


m 


CUBIC  TARDS  TO  MEAN  AREAS  FOJl  1OO  FEET  IN  LENGTH. 


M.A. 

•0 

•1 

•a 

•3 

•4 

•5 

•6 

•1 

•  8 

•9 

M.A. 

241 

892-593  892-963 

893-333 

893-704 

894074 

894-444 

894-815 

895-185 

895-556 

895-926  241 

242 

896-296  896-667 

897-037 

897-407 

897-778 

898-148 

898-519 

898-889 

899-259 

899-630  242 

243 

900-    900-370 

900-741 

901-111 

901481 

901-852 

902-222 

902-593 

902-963 

903-333  243 

214 

903-704  904-074 

904-444 

904-815 

905-188 

905-556 

905-926 

906-296 

900-667 

907-037 

244 

245 

907-407  907-778 

908-148 

908-519 

908-889 

909-259 

909-630 

910- 

910-370 

910-741 

245 

246 

911-111  911-4S1 

911852 

912-222 

912-593 

912-963 

913-333 

913-704 

914-074 

914-444 

246 

247 

914-815  915-185 

915-556 

915-926 

916-296 

916-667 

917-037 

917-407 

917-778 

918-148 

247 

24* 

918-519!  918-889 

919-259 

.919-630 

920- 

920-370 

920741 

921-111 

921-481 

921-852 

248 

249 

922-222 

922-593 

922-963 

923-3:33 

923704 

924-074 

924-444 

924-815 

925-185 

925-556 

249 

250 

925-926 

926-296 

926-667 

927-037 

927-407 

927-778 

928-148 

928519 

928-889 

929-259 

250 

251 

929-630 

930- 

930-370 

930-741 

931-111 

931-481 

931-852 

932-222 

932-593 

932-963 

251 

252 

933-333 

933-70-1 

9:34-1(74 

934-444 

9:34-815 

935-185 

935-556 

935-926 

936-296 

936-667 

252 

253 

937-037 

937-407 

937-778 

9.J8-148 

938  519 

938-889 

939-259 

939630 

940- 

940-370 

2f»3 

254 

940-741 

911-111 

941-481 

941-852 

942-222 

942593 

942-963 

943-333 

943-704 

944074 

254 

255 

944-444  944-815 

945-185 

945-556 

945-926 

946-296 

946-667 

947-037 

947-407 

947-778 

255 

256 

948-148  j  948-519 

948-889 

949259 

949-630 

950- 

950-370 

950-741 

951-111 

951-481 

256 

257 

951-852  i  952222 

952-693 

952-963 

953-333 

953704 

954  074 

954-444 

954815 

955-185 

257 

258 

955-556 

955-926 

956-296 

956  667 

957-037 

957-407 

957-778  958-148 

958-519 

958-889 

i>e  ,Q 

259 

959-259 

959-630 

960- 

960-370 

960-741 

96M11 

961-481 

961-852 

962-222 

962-593 

259 

260 

962-963 

963-333 

963-704 

964-074 

964-444 

964-815 

965-185 

965-556 

965926 

966-296 

260 

261 

966-667 

967-037 

967-407 

967-778 

968-148 

968519 

968-889 

969-259 

9R9-630 

970- 

261 

262 

970-370 

970-741 

971-111 

971-481 

971-852 

972-222 

972-593 

972963 

973333 

973-704 

262 

263 

974-074  974  111 

974-815 

975-185 

975-556 

975-926 

976-296 

676-667 

977-037 

977-407 

263 

264 

977-778  978148 

978-519 

978-889 

979-259 

979-630 

980- 

980-370 

980-741 

981-111 

264 

265 

981-481  981-852 

982-222 

982-593 

982-963 

933-333 

983-704 

9K4-074 

984-444 

984-815 

265 

266 

985-185  985-556 

98.V926 

986-296 

986-667 

9H7-037 

9S7407 

987778 

988-148 

988-519 

266 

267 

988-889  !  989-259 

989630 

990- 

990-370 

990-741 

991-111 

991-481 

991-852 

992-222 

267 

263 

992-593  992-963 

993-3:33 

993-704 

994-074 

994-444 

994-815 

995-185 

995-556 

995-926 

268 

269 

996-296'  996667 

997-037 

997-407 

997-778 

998-148 

998-519 

998889 

999-259 

999-630 

269 

270 

1000- 

1000-370 

1000-741 

001-111 

1001-481 

1001-852 

1002222 

1002-593 

1002-963 

1003-333 

270 

271 

1003-704 

1004-074 

1004-444 

1004-815 

1005-185 

1005-556 

1005-926 

1006-296 

1006-667 

1007-037 

271 

272 

1007-407 

1007-778 

1008-148 

1008-519 

1008-889 

1009-259 

1009-630 

1010- 

1010-370 

1010-741 

272 

273 

1011-111 

1011-481 

1011-852 

1012-222 

1012-593 

1012-963 

1013*333 

1013704 

1014-074 

1014-444 

273 

274 

1014815  1015-1N5 

1015556 

1015-926 

1016-296 

1016-667 

1017-037 

1017-407 

1017-778 

101S-148 

274 

275 

1018-51911018-889 

1019-259 

1019-630 

1020- 

102<)-37l 

1020-741 

1021-111 

1021-481 

1021-852 

275 

276 

1022-222 

1022-593 

1022-963 

1023-333 

1023-704 

1024-074 

1024-444 

1024-815 

102;')  -is: 

1025-556  276 

277 

1025-926 

1026-296 

1026-667 

1027-037 

1027-407 

1027-778 

1028-148 

1028-519 

1028-8f;9 

1029-259 

277 

278 

1029-630 

1030- 

1030-370 

1030-741 

1031-111 

1031  481  1031-852 

1032-222 

1032-593 

1032-963 

278 

279 

1033-333 

10:53-704 

1034-074 

1034-444 

1034-815 

1035-185 

1035-556 

1035-926 

1036-296 

1036-667 

279 

2SO 

1037-037 

1037-407 

1037-778 

1038-148 

1038-519 

1038-889 

1039-259 

1039-630 

1040- 

1040-370 

2SO 

281 

1040-741 

1041-111 

1041-481 

1041-852 

1042-222 

1042-593 

1042963 

1043-333 

1043-704 

1044-074 

281 

282 

1014444 

1044-815 

1045-185 

1045-556 

1045-926  1046-296 

1046-667 

1047-037 

1047-407 

1047-778 

282 

283 

1048-148 

1048-519 

1048-889 

1049-259 

1049-630  1050- 

ior,o-370 

1050-741 

1051-111 

1051-481 

283 

284 

1051-852  j  1052-222 

1052-593 

1052-963 

1053-3:33 

1053-704 

1054-074 

1054-444 

1054-815 

1055-1S5 

284 

285 

1055-556  1055-926 

I056-29d 

1056-667 

1057-037 

1057-407 

1057-778 

1058-148 

1058-519 

1058-889 

285 

286 

1059259 

1059-630 

1060- 

1060-370 

1060-741 

1061-111 

1061-481 

1061-852  1062-222 

1062-593 

286 

287 

1062-963 

1063-333 

1063704 

1064-074 

1064-444 

1064-815 

1065-185 

1005-556 

1065-926 

1066-296 

287 

288 

1066-667 

1067-037 

10.J7407 

1067-778 

106S-148 

1068-519  1068-889 

1069259 

1009-630  1070- 

288 

289 

1070-370 

1070-741 

1071-111 

1071-481 

1071-852 

1072-222 

1072-593  1072-963 

1073-333  1073-704 

289 

290 

1074-074 

1074-444 

1074-815 

1075-185 

1075-556 

1075-926 

1076-296 

1076667 

1077037 

1077-407 

290 

291 

1077-778 

1078-148 

1078-519 

1078-889 

1079-259 

1079-630 

1080- 

10*0-370 

1080-741 

108M11 

291 

292 

1081-481 

1081-852 

1082-222 

1082-593 

1082-963 

1083-333 

1083-704 

1084-074 

1084-444 

1084-815 

292 

293 

1085-185 

10H5-55b 

1085-926 

1086-296 

1086-667 

1087-037 

1087-407 

1087-778 

1088-148 

1088-519 

293 

294 

1088-889 

1089-259 

1089-630 

1090- 

1090-370 

1090-741 

1091-111 

1091-481 

1091-852 

1092-222 

294 

295 

1092-593 

1092-963  1093-333 

1093-704 

1094-074 

1094-444 

1094-815 

1095-185 

1095-550 

1095-926 

295 

296 

1096-296 

1096-667  1097-037 

1097-407 

1097-778 

1098-148 

1098-519 

1098-889 

1099-259 

1099-630 

296 

297 

1100- 

1100370  1100-741 

1101111 

1101-481 

1101-852 

1102-222 

1102-593 

1102-963 

1103-333 

297 

298 

1103-704 

1104  074  '1104-444 

1104-815 

1105-185 

1105-550 

1105-924 

1  106-296 

1100-667 

1107-037 

298 

299 

1107-407 

1107-77H1  1108-148 

110S-519 

1108-889 

1109-259 

1109-63 

1110- 

1110-37( 

1110-741 

299 

300 

1111-111 

1111-48111111-852 

1112222 

1112-593 

1112-963 

1113-333 

1113-704 

1114-074 

1114-444 

300 

M.A.|   '0 

•1 

•a 

•3 

•4 

•5 

•6 

•1 

•8 

•0 

M.A. 

MEAN  AREAS  241  to  3OO. 

178 


RULES   FOR    THE    MEASUREMENT   OF   EARTHWORKS. 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  1OO  FEET  IN  LENGTH. 


M.A. 

•0 

•1 

•2     «3 

•4 

•5 

•6 

•7 

•8 

•9 

M.A. 

301 

1114-815 

1115-185 

1115-556 

1115-926 

1116-296 

1116  667 

1117-037 

1117-407  1117-778 

1118-148 

301 

302 

1118519 

1118-889 

1119-259 

1119630 

1120- 

1120-370 

1120-741 

1121-111  1121-481 

1121-852 

302 

303 

1122-222 

1122593 

1122-963 

1123-333 

1123704 

1124-074 

1124-444 

1124-815 

1125-lbS 

1125-556 

303 

304 

1125-926 

1126296 

1126-667 

1127-037 

1127  407 

1127-778 

1128-148 

1128-519 

1128-889 

1129-259 

304 

305 

1129-630 

1130- 

1130370 

1130-741 

1131-111 

1131-481 

1131-852 

1132-222 

11  32-593 

1132-9113 

SOS 

306 

1133-33:5 

1133-704 

1134-074 

1134-444 

1134815 

1135-185 

1135-556 

1135-926 

1136-296 

1136-667 

306 

307 

1137-037 

1137407 

1137-778 

1138-148 

1138-519 

1138-889 

1139-259 

1139-630 

1140- 

1140-370 

307 

308 

1140  741|  1141-111 

1141-481 

1141-852 

1142-222 

1142-593 

1142-963 

1143-333 

1143-704 

1144-074 

308 

309 

1144-444  1144-815 

1145-1S5 

1145-556 

1145-926 

1146-296 

1146-667 

1147-037 

1147-407 

1147  778 

309 

310 

1148-148 

1148-519 

1148-889 

1149-259 

1149-630 

1150- 

1150-370 

1150-741 

1151-111 

1151-481 

310 

311 

1151-852 

1152-222 

1152-593 

1152-963 

1153-333 

1153-704 

1154-074 

1154-444 

1154-815 

1155-185 

311 

312 

1155-556 

1155-926 

1156-296 

1156-667 

1157-037 

1157-407 

1157-778 

1158-148 

1158-519 

115S-8S9 

312 

313 

1159259 

1159-630 

1160- 

11  60-370 

1160-741 

1161-111 

1161481 

1161-852 

1162-222 

1162-593 

313 

314 

1162-963  1163-333 

1163-704 

1164074 

1164-444 

1164-815 

1165-185 

1165-556 

1165-926 

1166-296 

314 

315 

1166-66711167-037 

1167-407 

1167-778 

1168-148 

1168-519 

1168-889 

1169-259 

1169-630 

1170- 

315 

316 

1170-370 

1170-741 

1171-111 

1171-481 

Ii7  1-852 

1172-222 

1172-593 

1172-963 

1173-333 

1173-704 

316 

317 

1174-074 

1174-444 

1174-815 

1175-185 

1175-556 

1175-926 

1176-296 

1176-667 

1177-037 

1177-407 

317 

318 

1177-778 

1178-148 

1178519 

178-889 

1179-259 

1179-630 

1180- 

11SO-370 

1180-741 

1181-111 

318 

319 

1181-481 

1181-852 

1182-222 

1182-593 

11829rt3 

1183-333 

1183-704 

1184074 

1184-444 

1184-815 

319 

320 

1185-185 

1185-55(3 

1185  926 

1186-296 

1186-667 

1187-037 

1187-407 

1187-778 

1188-148 

1188-519 

320 

321 

1188-889 

1189-259 

1189-630 

1190- 

1190-370 

1190-741 

1191-111 

1191-581 

1191-852 

1192-222 

321 

322 
323 

1192593 
1196-296 

1192-903 
1196-667 

1193-333 
1197-037 

1193-704 
1197-407 

1194-074 
1197-778 

1194-444 
1198-148 

1194-815 
1198-519 

1195-185  1195-556 
1198-889  1199-259 

1195-926 
1199-630 

322 
323 

324 

1200- 

1200-370 

1200-741 

1201-111 

1201-481 

1201-852 

1202-222 

1202-593)  1202-963 

1203-333 

324 

325 

1203-704 

1204-074 

1204-444 

1204-815 

1205-185 

1205-556 

1205-926 

1206-296 

1206-667 

1207-037 

325 

326 

1207-407 

1207-778 

1208-148 

1208-519 

1208-889 

1209-259 

1209-630 

1210- 

1210-370 

1210-741 

3-26 

327 

1211-111 

1211-481 

1211-852 

212222 

1212-593 

1212-963 

1213-333 

1213-704 

1214-074 

1214-444 

327 

328 

1214-815 

1215-185 

1215-556 

1215-926 

1216-296 

1216-667 

1217-037 

1217-407 

1217-778 

1218-148 

328 

329 

1218-519 

1218-889 

1219-259 

1219-630 

1220- 

1220-370 

1220-741 

1221-111 

1221-481 

1221-852 

329 

330 

1222-222 

1222-593 

1222-963 

1223-333 

1223-704 

1224-074 

1224-444 

1224-815 

1225-185 

1225-556 

330 

331 

1225-926 

1226-296 

1226-667 

1227-037 

1227-407 

1227-778 

1228-148 

1228-519 

1228-889 

1229-259 

331 

332 

1229-630 

1230- 

1230-370 

1230-741 

1231-111 

1231-481 

1231-852 

1232-222 

1232-593 

1232-963 

332 

333 

1233-333 

1233-704 

1234-074 

1234-444 

1234-815 

1235-185 

1235-556 

1235-926 

1236-296 

1236-667 

333 

334 

1237-037 

1237-407 

1237*778 

1238-148 

1238-519 

1238-889 

1239-259 

1239-630 

1240- 

1240-370 

334 

335 

1240-741 

1241-111 

1241-481 

1241-852 

1242-222 

1242-593 

1242-963 

124S-333 

1243-704 

1244074 

335 

336 

1244-444 

1244-815 

1245-185 

1245-556 

1245-926 

1246-296 

1246-667 

1247-037 

1247-407 

1247-778 

336 

337 

1248-148 

1248-519 

1248-889 

1249-259 

1249-630 

1250- 

1250-370 

1250-741 

1251-111 

1251-481 

337 

338 

1251-852 

1252222 

1252-593 

1252-963 

1253-333 

1253-704 

1254-074 

1254-444 

1254-815 

1255-185 

338 

339 

1255-556 

1255926 

1256-296 

1256-667 

1257-037 

1257-407 

1257-778 

1258-148 

1258-519 

1258-889 

339 

340 

1259-259 

1259-630 

1260- 

1260-370 

1260-741 

1261-111 

1261-481 

1261-852 

1262-222 

1262-593 

340 

341 

1262-963 

1263-333 

1263-704 

1264-074 

1264-444 

1264-815 

1265-185 

1265-556 

1265-926 

1266-296 

341 

342 

1266-667 

1267-037 

1267-407 

1267-778 

1268-148 

1268-519 

1268-889 

1269-259 

1269-630 

1270- 

342 

343 

1270-370 

1270-741 

1271-111 

1271-481 

1271-852 

1272-222 

1272-593 

1272-963 

1273-333 

1273-704 

343 

314 

1274-074 

1274-444 

1274-815 

1275-185 

1275-556 

1275-926 

1276-296 

1276-667 

1277-037 

1277-407 

344 

345 

1277-778 

1278-148 

1278-519 

1278-889 

1279-259 

1279630 

1280- 

1280-370 

1280-741 

1281-111 

345 

346 

1281-481 

1281-852 

1282-222 

1282-593 

1282-963 

1283-333 

1283-704 

1284-074 

1284-444 

1284-815 

346 

347 

1285-185 

1285-556 

1285-926 

1286-296 

1286-667 

1287-037 

1287-407 

1287-778 

1288-148 

1288-519 

347 

348  1288-889 

1289-259 

1289-630 

1290- 

1290-370 

1290-741 

1291-111 

1291-481 

1291-852 

1292-222 

348 

349 

1292-593 

1292-963 

1293-333 

1293-704 

1294-074 

1294-444 

1294-815 

1295-185 

1295-556 

1295-926 

349 

350 

1296-296 

1296-667 

1297-037 

1297-407 

1297-778 

1298-148 

1298-519 

1298-889 

1299-259 

1299-630 

350 

351 

1300- 

1300370 

1300-741 

1301-111 

1301-481 

1301-852 

1302-222 

1302-593 

1302-963 

1303-333 

351 

352 

1303-704 

1304-074 

1304-444 

1304-815 

1305-185 

1305-556 

1305-926 

1306-296 

1306-667 

1307-037 

352 

353 

1307-407 

1307-778 

1308-148 

1308-519 

1308-889 

1309-259 

1309-630 

1310- 

1310-370 

1310-741 

353 

354 

1311-111 

1311-481 

1311-852 

1312-222 

1312-593 

1312-963 

1313-333 

1313-704 

1314-074 

1314-444 

354 

355 

1314-815 

1315-185 

1315-556 

1315-926 

1316-296 

1316-667 

1317-037 

1317-407 

1317-778 

1318-148 

355 

356 

1318-519 

1318-889 

1319-259 

1319-630 

1320- 

1320-370 

1320-741 

1321-111 

1321-481 

1321-852 

356 

357 

1322-222 

1322-593 

1322-963 

1323-333 

1323-704 

1324-074 

1324-444 

1324-815 

1325-185 

1325-556 

357 

358 

1325-926 

1326-296 

1326-667 

1327-037 

1327-407 

1327-778 

1328-148 

1328-519 

•1328-889 

1329-259 

358 

359 

1329-630 

1330- 

1330-370 

1330-741 

13X1-111 

1331-481 

1331-852 

1332-222 

1332-593 

1332-963 

359 

360 

1333-333 

1333-704 

1334-074 

1334-444 

1334-815 

1335-185 

1335-556 

1335-926 

1336-296 

1336-667 

360 

M.A. 

•O 

•1 

•a 

•3 

•4: 

•5 

•6 

•7 

•  8 

•9 

M.A. 

MEAN  AREAS  3O1  to  36O. 

RULES   FOR   THE   MEASUREMENT   OF    EARTHWORKS. 


179 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  1OO  FEET  IN  LENGTH. 


M.A. 

•0 

•1 

•a 

•3 

•4: 

•5 

•6 

•7 

•8 

•9 

M.A 

361 

1337-037  1337-407 

1337-778 

1338-148 

1338-519 

1338-889 

1339259 

1339-630 

1340- 

1340-370 

361 

362 

1340-741  11341-111 

1341481 

1341-852 

1342-222 

1342-593 

1342-WJ:! 

1343-333 

1343-704 

1344-074 

362 

363 

1344-444  1344-815 

1345-185 

1345-556 

1345-926 

1346-296 

1346-667 

1347-037 

1347-407 

1347-778 

363 

304 

1348-148  1348-519 

1348-889 

1349-259 

1349-630 

1350- 

1350-370 

1350-741 

1331-111 

1351-481 

364 

365 

1351-852  1  352-2-22 

1352-593 

1352-963 

1353-333 

1353-704 

1354-074 

1354-444 

1354-815 

1355-185 

365 

366 

1355-556  1355-926 

1356-296 

1356667 

1357-037 

1357-407 

1357778 

1358-148 

1358  519 

1358-889 

366 

367 

1359-259  1359-630 

1360- 

1360-370 

1360741 

1361-111 

1361481 

1361-852 

1362-222 

1362593 

367 

368 

1362-963 

1303-333 

1363-704 

1364-074 

1364-444 

1364-815 

1365185 

1365-556 

1365926 

1366-296 

368 

369 

1366-667 

1367-037 

1367-407 

1367-778 

I3n8-148 

1368519 

1368-889 

1369259 

1369-630 

1370- 

369 

370 

1370-370 

1370-741 

1371-111 

1371-481 

1371-852 

1372-222 

1372-593 

1372963 

1373-333 

1373-704 

370 

371 

1374-074 

1374-444 

1374-815 

1375-185 

1375-556 

1375-926 

1376-296 

1376-667 

1377-037 

1377-407 

371 

372 

1377778 

1378-148 

1378-519 

1378.8S9 

1379-259 

1379630 

1380- 

1380-370 

1380-741 

1381-111 

372 

373 

1381-481 

1381-852 

13V2-222 

1382-593 

1382-963 

1383-333 

1383-704 

1384-074 

1384-444 

1384-815 

373 

374 

1385-185 

1385-556 

1385-926 

1386-296 

1386-667 

1387-037 

387-407 

1387-778 

1388148 

1388-519 

374 

375 

138S-889  1389-259 

1389-630 

1390- 

1390-370 

1390-741 

1391-111 

1391-481 

1391-852 

1392-222 

375 

376 

1392-593 

L3921KI8 

1393-333 

1393-70* 

1394-074 

1394-444 

1394-815 

1395185 

1395-556 

1395-926 

376 

377 

1396-296 

1396-667 

1397-037 

1397-407 

1397-778 

1398-148 

139S-519 

1398-889 

1399-259 

1399-630 

377 

378 

1400- 

1400-370 

1400-741 

1401-111 

1401-481 

1401-852 

1402-222 

1402-593 

1402-963 

1403-333 

378 

379 

1403-704 

1404-074 

1404-444 

1404-815 

1  405-  IK.", 

1405-556 

1405-926 

1406-296 

1406-667 

1407-037 

379 

380 

1407-407 

1407-778 

1408-148 

1408-519 

1408-889 

1409-259 

1409-630 

1410- 

1410-370 

1410-741 

380 

381 

1411-111 

1411-481 

1411-852 

1412-222 

1412-593 

1412-963 

1413-333 

1413-704 

1414-074 

1414-444 

381 

382 

1414-815 

1415-185 

1415-556 

1415-926 

1416-296 

1416-667 

1417-037 

1417-407 

1417-778 

1418-148 

382 

3X3 

1418-519 

1418-889 

1419-259 

1419-630 

1420- 

1420-370 

1420-741 

1421-111 

1421-481 

1421-852 

383 

384 

1422-222 

1422-593 

1422-963 

1423-333 

1423-704 

1424074 

1424-444 

1424-815 

1425-185 

1425-556 

384 

;>>*;> 

1425-926 

1426-296 

1426-667 

1427-037 

1427-407 

1427-778 

1428-148 

1428-519 

1428-889 

1429-259 

385 

386 

1429-630 

1430- 

1430370 

1430-741 

1431-111 

1431-481 

1431  852 

1432-222 

1432-593 

1432963 

386 

387 

14313-333 

1433704 

1434074 

1434-444 

1434-815 

1435-185 

1435-556 

1435-926 

1436-296 

1436-667 

387 

388 

1437-037 

1437407 

1437-778 

1438-148 

1438-519 

1438-889 

1439-259 

1439-630 

1440- 

1440-370 

388 

3S9 

1440-741 

1441-111 

1441-481 

1441-852 

1442-222 

1442-593 

1442-963 

1443-333 

1443-704 

1444-074 

389 

390 

1444-444 

1444-815 

1445-185 

1445-556 

1445926 

1446296 

1446667 

1447-037 

1447-407 

1447-778 

390 

301 

1448-148 

1448-519 

1448:889 

1449-259 

1449-630 

1450- 

1450-370 

1450-741 

1451-111 

1451-481 

391 

392 

1451-852 

1468*828 

1452-593 

1452-963 

1453-333 

1453-704 

1454-074 

1454-444 

1454-815 

1455-185 

592 

393 

1455-556 

1455-926 

1456296 

1456-667 

1457*039 

1457-407 

1457-77S 

1458-148 

1458-519 

1458889 

393 

394 

1459-259 

1459-630 

1400- 

1460-370 

1460-741 

1461-111 

1461-481 

1461-852 

1462-222 

1462-593 

394 

899 

1462-963 

1463-333 

1463-704 

1464-074 

1464-444 

1464-815 

1465-185 

1465-556 

1465-926 

1466-296 

395 

396 

1466-667 

1467-037 

1467-407 

1467-778 

1468148 

1468-519 

1468-889 

1469-259 

1469-630 

1470- 

396 

397 

1470-370 

1470-741 

1471-111 

1471-481 

1471-852 

1472-222 

1  472-59:1 

1472-903 

1473-333 

1473-704 

397 

398 

1474-074 

1474-444 

1474-815 

1475-185 

1475-556 

1475-926  1476-296 

1476-6G7 

1477-037 

1477-407 

398 

399 

1477-778 

1478-148 

1478-519 

1478-889 

1479-259 

1479-630  11480- 

1480-370 

1480-741 

1481-111 

399 

400 

1481-481 

1481-852 

1482-222 

1482-593 

1482-963 

1483-333 

1483-704 

1484-074 

1484-444 

1484-815 

400 

401 

1485-185 

1485-556 

1485-926 

1486-296 

1486-667 

1487-037 

1487-407 

1487-778 

1488-148 

1488-519 

401 

402 

1488-889 

1489-259 

1489-630 

1490- 

1490-370 

1490-741 

1491-111 

1491-481 

1491-862 

1492-222 

402 

403 

1492-593 

1492-903 

1493-333 

1493-704 

1494-074 

1494-444 

1494-815 

1495-185 

1495-556 

1495-926 

403 

404 

1496-296 

1496-667 

1497-037 

1497-407 

1497-778 

1498-148 

U  98-51  9 

1498-889 

1499-259 

1499-630 

404 

405 

1500- 

1500-370 

1500-741 

1501-111 

1501-481 

1501-852 

1502-222 

1502-593 

1502-963 

1503-333 

405 

406 

1503-704 

1504-074 

1504-444 

1504-815 

1505-185 

1505-556 

1505-926 

1506-296 

1506-667 

1507-037 

406 

407 

1507-407 

1507778 

1508-  14S 

1508-519 

1508-889 

1509-259 

1509-630 

1510- 

1610-370 

1510-741 

407 

408 

1511-111 

1511-481 

1511-852 

1512-222 

1512-593 

1512-963 

1513-333" 

1513-704 

1614-074 

1514444 

408 

409 

1514-815 

1515-185 

1515-556 

1515  92C 

1516-296 

1516-667 

1517-037 

1517-407 

1517-778 

1518-148 

409 

410 

1518-519 

1518-889 

1519-259 

1519-630 

1520- 

1520-370 

1520-741 

1521-111 

1521-481 

1621-852 

410 

411 

1522-222 

1522-593 

1522-963 

1523-333 

1523-704 

1524-074 

1524-444 

1524-815 

1525-185 

1525-656 

411 

412 

1525-926 

1526-296 

1526-667 

1527-037 

1527-407 

1527-778 

1528-148 

1528-519 

1528-889 

1529-259 

412 

413 

1529-630 

1530- 

1530-370 

1530-741 

1531-111 

1531-481 

15:;l-852 

1532-222 

1532-593 

1532-963 

413 

414 

1533-333 

1533-704 

1534074 

1534-444 

1534-815 

1535-185 

1535-556 

1535-926 

1536-296 

1536-667 

414 

415 

1537-037 

1537-407 

1537-778 

1538-148 

1538-519 

1538-889 

1539-259 

1539-630 

1540- 

1540-370 

415 

416 

1540-741 

1541-111 

1541-481 

1541-852 

1542-222 

1542-593 

1542-963 

1543-333 

1543-704 

1544-074 

416 

417 

1544-444 

1544-815 

1545-185 

1545-556 

1545-926 

1546-296 

1546667 

1547-037 

1547-407 

1547-778 

417 

418 

1548-14* 

1548-619 

1548-889 

1549-259 

1549-630 

1550- 

1550-370 

1550-741 

1.55M11 

1551-481 

418 

419 

1551-852 

1  552-  222  1552-593 

1fM2-%3 

1553-333 

1553-704 

1554-074|  1554-444 

1554-815 

1555-185 

419 

420 

155o-a56 

ir,55-02rt  l.r).->6-2'J6 

1556667 

1557-037 

1557-407 

1557-778 

1558-148 

1558-519 

1558-889 

420 

M.A. 

•0 

•1 

•a 

•3 

•4 

•5 

•6 

•7 

•8 

•9 

M.A. 

MEAN  AREAS  361  to  42O. 

180 


RULES    FOR    THE    MEASUREMENT    OF    EARTHWORKS. 


CUBIC  YARDS  TO  MEAN  AREAS  J  OR  1OO  FEET  IN  LENGTH. 


M.A. 

•0   |   .i 

•a 

•3   1   •* 

•5 

•6 

•7 

•8 

•9 

l.A. 

421 

559-259  1559  630 

1560- 

1560-37011560-741 

1561-111 

561-481 

1561-852  !  1562-222 

562-593 

421 

422 

562-963  1563-333 

1563-704 

1564-074  11564-444 

1564-815 

1565-185 

565-55611565-926 

566-296 

422 

423 

566-667  11567-037 

1567-407 

1507-778  1568-148 

1568-519 

568-889 

569-259  '1569-630 

570- 

423 

424 

570  370  1  1570-741 

571-111 

1571-481 

1571-852 

1572-222 

1572-593 

1572-963 

1573-333 

573-704 

424 

425 

574-07411574-444 

574-815 

1575-185 

1575556 

1575-926 

576-296 

576-667 

1577-037 

577-407 

425 

426 

577-778  1578-148 

1578-519 

1578-889!  1579-259 

1579-630 

1580- 

1580-370  1580-741 

581-111 

426 

427 

581-481  1581-852 

5S2-222 

1582-593  1582-963 

1583-333 

1583-704 

584-074 

1584-444 

584-815 

427 

428 

585-185  1585-556 

585-926 

1586-296  1586-667 

1587-037 

1587-407 

1587-778 

1588-148 

1588-519 

428 

429 

588-889  1589-259 

589-630 

1590- 

1590-370 

1590-741 

1591-111 

1591-481 

1591-852 

1592-222 

429 

430 

592-593 

1592-963 

1593-333 

1593-704 

1594-074 

1594-444 

1594-815 

1595-185 

1595-556 

1595-926 

430 

431 

596296 

1596-667 

597-037 

597-407 

1597-778 

1  598-1  4S 

1598-519 

1598-889 

1599-259 

1599-630 

431 

432 
433 

600-    1600-370 
(303-704  1604-074 

600-741 
604-444 

601-111 
604-815 

1001-481 
1605-185 

1601-852 
1605-556 

1602-222 
1605-926 

1602-593  1602-963 
1606-296  1606-667 

1603-333 
607-037 

432 
433 

434 

607-407  j  1607-778 

1608-148 

608-519 

1608-889 

1609-259 

1609630 

1610- 

1610-370 

1610-741 

434 

435 

611111 

1611-481 

1611-852 

612-222 

1612-593 

1612-963 

1613-333 

1613-704 

1614-074 

1614-444 

435 

436 

614-815 

1615-185 

615-556 

615-926 

1616-296 

1616-667 

K17-037 

1617-407 

1617-778 

1618-148 

436 

437 

618-519 

1618-889 

619-259 

619  630 

1620- 

1620-370 

1620-741 

1621-111 

1621-481 

1621-852 

437 

438 

622-222 

1622-593 

1622-963 

623-333 

1623-704 

1624-074 

1C24-444 

1624-815 

1625-185 

1625-556 

438 

439 

625-926 

1626-296 

1626-667 

627-037 

1627-407 

1627-778 

1628-148 

1628-519 

1628-889 

1629-259 

439 

440 

1629-630 

1630- 

1630-370 

630-741 

1631-111 

1631-481 

1631-852 

1632-222 

1632-593 

1632-963 

440 

441 

1633-333 

1633-704 

1634-074 

634-444 

1634-815 

1635-185 

1635-556 

1635-926 

1636-296 

1636-667 

441 

442 

1637-007 

1637-407 

637-778 

638-148 

1638-519 

1638-889 

1639-259 

1639-630 

1640- 

1640-370 

442 

443 

1640-741 

1641-111 

1641-481 

641-852 

1642-222 

1642-593 

1642-963 

1643-333 

1643-704 

1644-074 

443 

444 

1644-444 

1644-815 

.645-185 

645-556 

1645-926 

1646-296 

1646-667 

1647-037 

1647-407 

1647-778 

444 

445 

1648-148 

1648-519 

1648-889 

649259 

1649-630 

1650- 

1650-370 

1650-741 

1651-111 

1651-481 

445 

446 

1651-852 

1652-222 

1652-593 

652-963 

1653-333 

1653-704 

1654-074 

1654-444 

1654-815 

1655-185 

446 

447 

1655-556 

1655-926 

1656-296 

656-667  1657-037 

1657-407 

1657-778 

1658-148 

1658-519 

1658-889 

447 

448 

1659-259 

1659-630 

1660- 

1660-370'  1660-741 

1661-111 

1661-481 

1661-852 

1662-222 

1662-593 

448 

449 

1662-963 

1663-333 

1663-704 

1664-074 

1664-444 

1664-815 

1665-185 

1665-556 

1665-926 

1666-296 

449 

450 

1666-667 

1667-037 

1667-407 

1667-778 

1668-148 

1668-519 

1668-889 

1669-259 

1669-630 

1670- 

450 

4-31 

1670-370 

1670-741 

1671-111 

1671-481 

1671-852 

1672-222 

1672-593 

1672-963 

1673-333 

1673-704 

451 

452 

1674-074 

1674-444 

1674-815 

1675-185 

1675-550 

1675-926 

1676-296 

1676-667 

1677-037 

1677-407 

452 

453 

1677-778 

1678-148 

'1678-519 

1678-889 

1679-259 

1679-630 

1680- 

1680-370 

1680-741 

1681-111 

453 

454 

1681-481 

1681-852 

1682-222 

1682-593 

1682-963 

1683-333 

1683-704 

1684-074 

1  684-444 

1684-815 

454 

455 

1685-185 

1685-556 

1685-926 

1686-296 

1686-667 

1687-037 

1687-407 

1687-778 

1688-148 

1688-519 

455 

456 

1688-889 

1689-259 

1689-630 

L690- 

1690-370 

1690-741 

1691-111 

1691-481 

1691-852 

1692-222 

456 

457 

1692-593 

1692-963 

1693-333 

1693-704 

1694-074 

1694-444 

1694-815 

1695-185 

1695-556 

1695-926 

457 

458 

1696-296 

1696-667 

1697-037 

1697-407 

1697-778 

1698-148 

1698-519 

1698-889 

1699-259 

1699-630 

458 

459 

1700- 

1700-370 

1700-741 

1701-111 

1701-481 

1701-852 

1702-222 

1702-593 

1702-963 

1703-333 

459 

460 

1703-704 

1704-074 

1704-444 

1704-815 

1705-185 

1705-556 

1705-920 

1706-296 

1706-667 

1707-037 

460 

461 

1707-407 

1707-778 

1708-148 

1708-519 

1708-889 

1709-259 

1709-630 

1710- 

1710-370 

1710-741 

461 

462 

1711-111 

1711-481 

1711-852 

1712-222 

1712-593 

1712-96b 

1713-333 

1713-704 

1714-074 

1714-444 

462 

463 

1714-815 

1715-185 

1715-556 

1715-926 

1716-296 

1716-667 

1717-037 

1717-407 

1717-778 

1718-148 

463 

464 

465 

1718-519 
1722-222 

1718-889 
1722-593 

1719-259 
1722-963 

1719-630 
1723333 

1720- 
1723-704 

1720-370  1720-741 
1724-074;  1724-444 

1721-111 
1724-815 

1721-481 
1725-185 

1721-852 
1725-556 

464 
465 

466 

1725-926 

1726-296 

1726-667 

1727-037 

1727-407 

1727-778 

1728-148 

1728-519 

1728-889 

1729-259 

406 

467 

1729-630 

1730- 

1730-370 

1730-741 

1731-111 

1731-481 

1731-852 

1732-222 

1732-593 

1732-963 

467 

468 

1733-333 

1733704 

1734-074 

1734-444 

1734-815 

1735-185 

1735-556 

1735-926 

1736-296 

1736-667 

468 

469 

1737-037 

1737-407 

1737-778 

1738-148 

1738-519 

1738-889 

1739-259 

1739-630 

1740- 

1740-370 

469 

470 

1740-741 

1741-111 

1741-481 

1741-852 

1742-222 

1742-593 

1742-963 

1743-333 

1743-704 

1744-074 

470 

471 

1744-444 

1744-815 

1745-185 

1745-556 

1745-926 

1746-296 

1746-667 

1747-037 

1747-407 

1747-778 

471 

472 

1748-148 

1748-519 

1748-889 

1749-259 

1749-630 

1750- 

1750-370 

1750-741 

1751-111 

1751-481 

472 

473 

1751-852 

1752-222 

1752-593 

1752-963 

1753-333 

1753-704 

1754-074 

1754-444 

1754-815 

1755-185 

473 

474 

1755-556 

1755-926 

1756-296 

1756-667 

1757-037 

1757-407 

1757-778 

1758-148 

1758-519 

1758-889 

474 

475 

1759-259 

1759-630 

1760- 

1760-370 

1760-741 

1761-111 

1761-481 

1761-852 

1762-222 

1762-593 

475 

476 

1762-963 

1763-333 

1763-704 

1764-074 

1764-444 

1764-815 

1765-185 

1765-556 

1765-926 

1766-296 

476 

477 

1766-667 

1767-037 

1767-407 

1767-778 

1768-148 

1768-519 

1768-889 

1769-259  11769-630 

1770- 

477 

478 

1770-370 

1770-741 

1771-111 

1771-481 

1771-852 

1772-222 

1772-593 

1772-963 

1773-333 

1773-704 

478 

479 

1774-074 

1774-444 

1774-815 

1775-185 

1775-556 

1775-926 

1776-296 

1776-667 

1777-037 

1777-40" 

479 

480 

1777-778 

1778-148 

1778-519 

1778-889 

1779-259 

1779-630 

1780- 

1780-370 

1780-741 

1781-111 

480 

M.A 

-O 

•1 

•a 

•3 

•4: 

•5 

•6 

•7 

•8 

•9 

M.A. 

MEAN  AREAS  421  to  48O. 

RULES    FOR   THE    MEASUREMENT    OF    EARTHWORKS. 


181 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  WO  FEET  IN  LEXGTIT. 


M.A. 

•0 

•1 

•2 

•3 

•4: 

•5 

•6 

•7 

•8 

.9  IM.A. 

481 

1781-481 

1781-852 

1782-222 

1782-593 

1782-963 

1783-333 

1783-704 

1784-074 

1784-444 

1784-815 

481 

482 

1785-185 

1785-556 

1785-926 

1786296 

1786-667 

1787-037 

1787-407 

1787-778 

1788-148 

1788-519 

482 

483 

1788-889 

1789-259 

1789630 

1790- 

1790-370 

1790-741 

1791-111 

1791-481 

1791-852 

1792-222 

483 

484 

1792-593 

1792-963 

1793-333 

1793-704 

1794-074 

1794444 

1794-815 

1795-185 

1795-556 

1795-926 

4^4 

485 

1796-296 

1796-667 

1797-037 

1797-407 

1797-778 

1798148 

1798-51911798-889 

1799-259 

1799-630 

485 

486 

1800- 

1800-370 

1800-741 

1801-111 

1801-481 

1801-852 

1802-222  1802-593 

1802  903 

1803-333 

486 

487 

1803-704 

1804-074 

1804-444 

1804-815 

1805-185 

1805-556 

1805926 

1806-296 

1806-607 

1807-037 

487 

488 

1807-407 

1807-778 

1808-148 

1808-519 

1808-889 

1809-259 

1809-630 

1810- 

1810-370 

1810-741 

488 

489 

1811-111 

1811-481 

1811-852 

1812-222 

1812-593 

1812-903 

1813-333 

1813-704 

1814-074 

1814444 

489 

490 

1814-815 

1815-185 

1815-556 

1815926 

1816-296 

1816-667 

1817-037 

1817-407 

1817-778 

1818-148 

490 

491 

1818-519 

1818-889 

1819-259 

181963ft 

1820- 

1820-370 

1820-741 

1821-111 

1821-481 

1821-852 

491 

492 

1822-222|  1822-593 

18-22-963 

1823333 

1823-704 

1824-074 

1824-444 

1824-815 

1825-185 

1825-556 

492 

493 

1825-926  11826-296 

1826-667 

1827-037 

1827-407 

1827  778 

1828-148 

1828-619 

1828-889 

1829-259 

493 

494 

1829-6301  1830- 

1830-370 

1830-741 

1831-11111831-481 

1831-852 

1832  222 

1832-593 

1832963 

494 

495 

1833-333  1833-704 

1834074 

1834-444 

1834-815  l-:;.vis;, 

1835-556 

1835-926 

1836-296 

1836  667 

495 

496 

1837-037  1837-407 

1837-778 

1838-148 

1838-519 

1838-88U 

1839-259 

1839-630 

1840- 

1840-370 

496 

497 

1840-741  '1841-111 

1841-481 

1841-852 

1842-222 

1842-593 

1842-963 

1843-333 

1843-704 

1844074 

4<»7 

498 

1844-444;1844-815 

1845-185 

1845-556 

1845-926 

1846296 

1846-667 

1847-037 

1847-407 

1847-778 

498 

499 

1848-148il848-519 

1848-889 

1849-259 

1849-630 

1850- 

1850-370 

1850-741 

1851111 

1851-481 

499 

500 

1851-852 

1852-222 

1852-593 

1852-963 

1853-333 

1853-704 

1854-074 

1854-444 

1864-815 

1855-185 

500 

501 

1856-660 

1855-926 

1856-296 

1856-667 

1857-037 

1857-407 

1857-778 

1868-148 

1858-519 

1858-889 

501 

502 

1859-259!  1859-630 

I860- 

1860-370 

1860-741 

1861-111 

1861-481 

1861-852 

1862-222 

1862-593 

502 

503 

1862-963  ,1863  333 

1863-704 

1864-074 

li-  64-444 

1864-815 

186*186 

1865-556 

1865-926 

1866-296 

503 

504 

1866-667;  1867  '037 

1867-407 

1867-778 

1868-148 

1868-519 

1868-889 

1869-259 

1869-630 

1870- 

504 

505 

187o-;j?i  1870741 

1871-111 

1871-481 

1871-852 

1872-222 

1872-593 

1872-9W 

1873-333 

1873-704 

505 

506 

1874-074  1874-444 

1874-815 

1876-185 

1875-556 

1875-926 

1876-296 

1876-667 

1877-037 

1877-407 

506 

507 

1877-778  1878-148 

187S-519 

1878-889 

1879259 

1879-630 

1880- 

1880-370 

1880-741 

1881-111 

507 

508 

1881-481  1881-852 

lss-2-j-j-j 

18S2-593 

ISBfrMfl 

is*:;-:;:;:' 

1883-704 

L-sH-74 

1  884-444 

1884815 

508 

509 

1885-185  1885-556 

1-85-926 

1886-296 

18*6-667 

1887-081 

1887-407 

1887-778 

1888-148 

1888-519 

509 

510 

1888-8S9 

1889-259 

1889-630 

1890- 

1890  370 

1890-741 

1891-111 

1891-481 

1891-852 

1892-222 

510 

511 

1892-593 

1892-963 

1893-333 

1803-704 

1894-074 

1894-444 

1894-815 

1895-185 

1895-556 

1895-926 

511 

612 

1896-296 

189*601 

1897-037 

1897-407 

1897-77-S 

1898-148 

1898-519 

1898-889 

1899-259 

1899-630 

512 

513 

1900- 

1900-370 

1900-741 

1901-111 

1901-481 

1901-852 

1902-222 

1902-593 

1902-963 

1903-333 

513 

514 

1903-704 

1904-074 

1  904-444 

1904-815 

1905-185 

1905-556 

1905-926 

1906-290 

1906-667 

1907-037 

514 

515 

1907-407 

1907-778 

1908-148 

1908-519 

1908-889 

1909-259 

1909-630 

1910- 

1910-370 

1910-741 

515 

510 

1911-111 

1911-481 

1911-852 

1912-222 

1912-593 

191'2-yrw 

1913-333 

1913-704 

1914-074 

1914-444 

516 

517 

1914-815 

1915-185 

1915-556 

1915-926 

1916296 

1916-667 

1917-037 

1917-407 

1917-778 

1918-148 

517 

518 

1918-519 

1918-889 

1919-259 

1919-630 

1920- 

1920-370 

1920-741 

1921-111 

1921-481 

1921-852 

518 

519 

1922-223 

1922-593 

1922963 

1923-333 

19-23704 

1924-074 

1924-444 

1924815 

1925-185 

1925-556 

519 

520 

1925-926 

1926-296 

1926-667 

1927-037 

1927-407 

1927-778 

1928-148 

1928-519 

1928-889 

1929-259 

520 

521 

1929-630 

1930- 

1930-370 

1930-741 

1931-111 

1931-481 

1931-852 

1932-222 

1932-593 

1932963 

521 

522 

1933333 

1933-704 

1934-074 

1934-441 

1934-815 

1935-185 

1935-556 

1935-926 

1936-296 

1936-667 

522 

523 

1937-037 

1937-407 

1937-778 

1938-148 

1938-519 

1938-889 

1939-259 

1939-630 

1940- 

1940370 

523 

524 

1940-741 

1941-111 

1941-481 

1941-852 

1942-222 

1942-593 

1942-963 

1943-33:' 

1943-704 

1944-074 

524 

526 

1944-444 

1944-815 

1945185 

1945-556 

1945-926 

1946-296 

1946-667 

1947-037 

1947-407 

1947-778 

tor. 

526 

1948-148 

1948-519 

1948-889 

1949-259 

1949-G30 

1950- 

1950-370 

1950-741 

1951-111 

1951  481 

526 

527 

1951-852 

1952-222 

1952-593 

1952963 

1953-333 

1953-704 

1954-074 

1954-444 

1954-815 

1955185 

5-27. 

528 

1955-556 

1955-926 

1956-296 

1956-667 

1957-037 

1057-407 

1957-778 

1958-148 

1958-519 

1958-889 

528 

529 

1959-259 

1959-630 

I960- 

1960-370 

1960-741 

1961-111 

1961-481 

1961-852 

1962-222 

1962-593 

529 

530 

1962963 

1963333 

1963-704 

1964-074 

1964-444 

1964-815 

1965-185 

1965-556 

1965-920 

1966-296 

530 

531 

1966-667 

1967-037 

1967-407 

1967-778 

1968-148 

1968-519 

1968-889 

1969-259 

1969-630 

1970- 

531 

532 

1970-370 

1970-741 

1971-111 

1971481 

1971-852 

1972-222 

1972-593 

1972963 

1973333 

1973-704 

532 

533 

1974-074 

1974-444 

1974-815 

1975-185 

1975-556 

1975-920 

1976-296 

1976-667 

1977-037 

1977-407 

533 

:>34 

1977-778 

1978-148 

1978-519 

1978-889 

1979-259 

1979-630 

1980- 

1980-370 

1980-741 

1981-111 

534 

535 

1981-481 

1981-852 

1982-222 

1982-593 

1982-963 

1983-333 

1983-704 

1984-074 

1984-444 

1984-815 

535 

536 

1985-185 

1985-556 

1  985-926 

1986-296 

1986667 

1987-037 

1987-407 

1987-778 

19*8-148 

1988-519 

536 

537 

1988-889 

1989-259 

1989-630 

1990- 

1990370 

1990-741 

1991-111 

1991-481 

1991-852 

1992-222 

537 

538 

1992-593 

1992-963 

1993-333 

1993-704 

1994-074 

1994-444 

1994-815 

1995-185 

1995-556 

1995-926 

538  , 

:>39 

1996-296 

1996-667 

1997-037 

1997-407 

1997*778 

1998-148 

199S-519 

1998-889 

1999-259 

1999630 

539  ' 

540 

2000- 

2000-370 

2000-741 

2001-111 

2001-481 

2001-852 

2002-222 

2002-593 

2002-963 

2003-333 

540  ; 

M.A. 

•O 

•  1 

•a 

•3 

•4: 

-5 

-0 

•7 

•8 

•9 

M.A. 

ME  A  If  AREAS  48  1  to  54Q. 

182 


RULES    FOR    THE    MEASUREMENT    OF    EARTHWORKS. 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  WO  FEET  IN  LENGTH. 


M.A. 

•0 

•  1 

•% 

•3 

•4 

•5 

•6 

•7 

•8 

•9 

M.A. 

541 

J003-704  2004-074 

2004-444 

004-815 

2005-185 

2005-556 

2005-926 

006-296 

2006-607 

2007-037 

541 

542 

2007-407,  2007-778 

2008-148 

008-519 

008-889 

2U09-259 

2009-630 

010- 

2010-370 

2010-741 

542 

543 

2011-111  2011-481 

2011-852 

012-222 

2012-593 

2012-963 

2013-333 

013-704 

2014-074 

2014444 

543 

544 

2014-815;  2015-  185 

2015-556 

015-926 

2016-296 

2016-667 

2017-037 

017-407 

2017-778 

2018-148 

544 

545 

2018-519  2018-889 

2019-259 

019-630 

2020- 

2020-370 

2020-741 

021-111 

2021-481 

2021-852 

545 

546 

2022-222  2022-593 

2022-963 

023-333 

2023-704 

2024-074 

2024-444 

024-815 

2025-185 

2025-550 

546 

547 

2025-926  2026-296 

2026-667 

027-037 

2027-407 

2027-778 

2028-148 

028-519  2028-889 

2029-259 

547 

518 

2029-030,2030- 

2030-370 

030-741 

2031-111 

2031-481 

2031-852 

2032-222  12032-593 

2032-963 

548 

549 

2033  333  2033-704 

2034-074 

034-444 

2034-815 

2035-185 

2035-55S 

2035-926  2036-290 

2036-667 

549 

550 

2037-037 

2037-407 

2037-778 

038-148 

2038-519 

2038-889 

2039-259 

2039-630 

2040- 

2040-370 

550 

551 

2040-741 

2041-111 

2041-481 

041-852 

2042-222 

•2042-593 

2042-963 

2043-333 

2043704 

044-074 

551 

5.V2 

2044-444  2044-815 

2045  185 

045-506 

2045-926 

2046-296 

2046-667 

2047-037 

2047-407 

047-778 

552 

553 

2048-148  2048-519 

2048-889 

049-259 

2049-630 

2050- 

2050-370 

2050-741 

2051-111 

051-481 

553 

554 

2051-852  2052-222 

2052-593 

052-963 

2053-333 

2053-704 

2054-074 

2054-444 

2054-815 

055-185 

554 

555 
556 

2055-556  12055-926 
2059-259  2059-630 

2056-296 
2060- 

056-667 
060-370 

2057-037 
2060-741 

057-407 
2061-111 

2057-778 
2061-481 

2058-148 
2061-852 

2058-519 
2062-222 

2058-889 
2062-593 

555 
556 

557 

2062-963 

2063-333 

2063-704 

064'07-i 

2064-444 

2064-815 

2065-1F5 

2065-550 

2065-926 

2000-290 

557 

558 

206(5-667 

2067-037 

2067-407 

2067-778 

2068-148 

2068-519 

2068-889 

2069-259 

2069-630 

2070- 

558 

559 

2070-370 

2070-741 

2071-111 

2071-481 

2071-852 

2072-222 

2072-593 

2072-96:5 

2073-333 

2073-704 

559 

500 

2074-074 

2074-444 

2074-815 

2075-185 

2075-556 

2075-926 

2076-296 

2076-667 

2077-037 

2077-407 

560 

561 

2077-778 

2078-148 

2078-519 

2078-889 

2079-259 

2079-630 

2080- 

2080-370 

2080-741 

2081-111 

501 

502 

2081-481 

2081-852 

2082-222 

2082-593 

2082  963 

2083-333 

2083-704 

2084-074 

2084-444 

2084-815 

562 

563 

2085-185 

2085-556 

2085-926 

2086-296 

2086-667 

2087-037 

2087-407 

20S7-778 

2088-148 

2088-519 

503 

564 

2088-889 

2089-259 

2089-630 

2090- 

2090-370 

2090  741 

2091-111 

2091-481 

2091-852 

2092-222 

504 

565 

2092-593 

2092-963 

2093-333 

2093-704 

2094-074 

2094-444 

2094-815 

•2095-185 

2095-556 

2095-920 

565 

5fiO 

2096-296 

2096-667 

2097-037 

2097-407 

2097-778 

2098-148 

2098-519 

2098-889 

2099-259 

2099-6:50 

566 

567 

2100- 

2100-370 

2100-741 

2101-111 

2101-481 

2101-852 

2102-222 

2102-593 

2102-963 

210:333:5 

567 

568 

2103-704 

2104-074 

2104-441 

2104-815 

2105-185 

2105-556 

2105-920 

2106-296 

2106-667 

2107-037 

508 

§69 

2107-407 

2107-778 

2108-148 

2108-519 

2108-889 

2109-259 

2109-6-H 

2110- 

2110-370 

2110-741 

509 

570 

2111-111 

2111-481 

2111-852 

2112-222 

2112-593 

2112963 

2113-333 

2113-704 

2114-074 

2114-444 

570 

571 

2114-815 

2115-185 

2115-556 

2115-926 

2116-296 

2116-C67 

2117-03" 

2117-407 

2117-778 

2118-148 

571 

572 

2118-519 

2118-869 

2119-259 

2119-630 

2120- 

2120-370 

2120-741 

2121-111 

2121-481 

2121-852 

572 

573 

2122-222 

2122-593 

•2122-963 

2123333 

2123-704 

2124-074 

•2124-444 

2124-815 

2125-185 

2125-556 

573 

574 

2125926 

2126-296 

2126-607 

2127-037 

2127-407 

2127-77* 

2128-148 

2128-519 

21  28-889 

212925< 

574 

575 

21-29-630 

2130- 

2130-370 

2130-741 

2131-111 

2131-481 

2131-852 

2132-222 

2132-593 

2132-963 

575 

576 

2133-333 

2133-704 

2134-074 

2134-444 

2134-815 

2135-18' 

2135-556 

2135-926 

2136-296 

2130-667 

576 

577 

2137-037 

2137-407 

2137-778 

2138-148 

2138-519 

2138-889 

21o9-259 

2139-630 

2140- 

2140-370 

577 

578 

2140-741 

2141-111 

2141-481 

2141-852 

2142-222 

2142-593 

2142-96! 

2143-333 

2143-704 

2144-074 

578 

579 

2144-444 

2144-815 

2145-185 

2145-556 

2145  926 

2146-290 

2i4(/667 

2147-037 

2147  407 

2147-778 

579 

580 

2148-148 

2148-519 

2148-889 

2149-259 

2149-630 

2150- 

2150-370 

2150-741 

2151-111 

2151-481 

580 

581 

2151-852 

2152-222 

2152-593 

2152-96.° 

2153-333 

2153-704 

2154-074 

2154-444 

2154-815 

2155-185 

581 

582 

2155-556 

2155-926 

2156-296 

2156-667 

2157-037 

2157-407 

2157-77 

2158-14S 

2158-519 

2158  889 

582 

583 

2159-259 

2159-630 

2160- 

2160-370 

2160-741 

2161-111 

2101-48 

2161-852 

2162-222 

2162-593 

583 

584 

2162-963 

2163-333 

2163-704 

2164-074 

2164-444 

2164-815 

2165-18 

2165-556 

2165-926 

2166-290 

584 

585 

21t)6-667 

2167-037 

2167-407 

2167-778 

2168-148 

2168-519 

2168-88 

2169-259 

2169-630 

2170- 

585 

586 

2170-370 

2170-741 

2171-111 

2171-481 

2171-852 

2172-222 

2172-59 

2172-963 

2173-333 

2173-704 

586 

587 

2174-074 

2174-444 

2174-815 

2175-185 

2175-556 

2175-926 

2176-29 

2176-667 

2177-037 

2177-40- 

587 

588  2177-778 

2178-148 

2178-519 

2178-889 

2179-259 

2179-630 

2180- 

2180-370 

2180-741 

2181-111 

588 

589 

2181-481 

2181-852 

21S2-222 

2182-593 

2l82-96o 

2183-333 

2183-70 

2184-074 

2184-444 

2184-815 

589 

590 

2185-185 

2185-556 

2185-926 

2186296 

2186-667 

2187-037 

2187-40 

2187-778 

2188-148 

2188-519 

590 

591 

2188-889 

2189-259 

2189-630 

2190- 

2190-370 

2190-741 

2191-11 

2191-481 

2191-852 

2192-222 

591 

592 

2192-593 

2192-963 

2193-33: 

2193704 

2194-074 

2194-444 

2194-81 

2195-185 

2195-55t) 

2195-926 

592 

593 

2196-296 

2196-667 

2  197  -03" 

2197-4(7 

2197778 

2198-148 

2198-61 

2198-889 

2199-259 

2199-631 

593 

594 

2200- 

2200-37C 

2200-741 

2201-111 

2201-481 

2201852 

2202-22 

2202-593 

2202-963 

2202-3:33 

594 

595 

2203-704 

2204-074 

2204-444 

2204-815 

2205-185 

2205-556 

2205-92 

2206-296 

2206-667 

2207-037 

595 

596 

2207-407 

2207-77* 

2208-148 

2208-519 

-'208-889 

2209-259 

2209-63 

2210- 

2210-370 

2210-741 

596 

597 

2211-111 

2211-481 

2211-852 

2212-222 

2212-593 

2212  9^3 

2213-33 

2213-704 

2214-074 

2214-444 

597 

598 

2214815 

2215-185 

2215-556 

•2215-92C 

2216-296 

2216-667 

2217-03 

2217-407!  2217-778 

2218-148 

598 

599 

2218-519 

2218-88P 

2219-259 

2219-630 

•2220- 

2-J20-370 

2220-74 

2221-111 

2221-481 

2221-852 

599 

600 

2222-222 

•2222-593 

2222-96: 

2223-333 

2223-704 

2224-074 

2224-44 

2224-815 

2225-185 

2225-556 

6UO 

M.A 

•O 

•1 

•a 

•3 

•4: 

•5 

•6 

•7 

•8 

•9 

M.A. 

MEAN  AREAS  54  1  to  6OO. 

RULES   FOR   THE    MEASUREMENT   OF   EARTHWORKS. 


183 


CVTtIC  YARDS  TO  MEAN  AREAS  FOR  1OO  FEET  IN  LENGTH. 


M.A. 

•0 

•  1 

•2 

•3 

•4: 

•5 

•6 

•7 

•8 

•9 

M.A. 

001 

2225-  92« 

2220-296 

2220-667 

22-27-037 

2227-407 

2227778 

2228-148 

2228-519 

2228-889 

2229-259 

601 

602 

2229-630 

2230- 

•22::o-370 

2230-741 

2231  111 

2231-481 

223  1  852 

2232*222 

2232-593 

2232-903 

602 

003 

2233-333 

2233-704 

2234-074 

22»4-444 

2234-815 

2235-185 

2235-550 

2235-92G 

2236-296 

2230-067 

603 

604 

22-37-037 

22.57-407 

2237-778 

2238-148 

2238-519 

2238-889 

2239-259 

22i>9-t30 

2240- 

2240370 

604 

HI  )5 

2240-741 

2241-111 

2241-481 

2241-852 

2242222 

2242-593 

^242-903 

2243-333 

2243-704 

2244-074 

605 

666 

2J44-444 

2244-815 

2245-  1S5 

2245-556 

2245  920 

2246-296 

2240-667 

2247-C37 

2247-407 

2247-778 

606 

f»7 

2248-14S 

2243-519 

224S-889 

2249-259 

2249-030 

2250- 

2250-370 

2250-741 

2251-111 

2251-481 

607 

('08 

2''5l-85"> 

2252-222 

J252-593 

2252-963 

2253-333 

2253-704 

2254-074 

2254-444 

2254-815 

2255-185 

608 

0)9 

225.)  r,5G 

2255-920 

2250-296 

2-256-607 

2257-037 

2257-407 

2257-778 

2258-148 

2258-519 

2258-889 

609 

tilO 

2259-253 

2259-630 

2260- 

2260-370 

2260-741 

2261111 

2261-481 

2261-852 

2262-222 

2262-593 

610 

611 

22^2-963 

2263-333 

2263-704 

22R4-074 

2264-444 

2-264-815 

2265-185 

2265-556 

22P5-920 

2266-296 

611 

012 

2  JGtt  667 

2267037 

2207-407 

2207-778 

22G8-14S 

226S-519 

2268-8*9 

22C9-259 

2269-630 

2270- 

612 

B13 

2270-370 

2270-741 

2271-11! 

2271-481 

2271-852 

2272-222 

2272-593 

•2272-963 

2273-33o 

2273704 

613 

014 

2274-074 

2274-441 

2274-815 

2275-185 

2275-556 

2275-920 

2-276-296 

227G-C67 

2277-037 

2277-407 

614 

015 

2277-778 

2278-148 

2278519 

227S-8S9 

2279-259 

•2279-630 

2280- 

2-280-37C 

2280-741 

2281-111 

615 

616 

2281-481 

22S1-852 

2282-'222 

2282-593 

2282-9H3 

2283-333 

2283704 

2284-074 

2284-444 

22*4-815 

616 

617 

2285-185 

22S5-55G 

22S5-92G 

2286-296 

22SG-C67 

2287  037 

2287-407 

2287-778 

2288-148 

2288-519 

617 

618 

228H-889 

'2283-259 

2289-630 

2230- 

2290-370 

2290-741 

2291-111 

2291-481 

•2291-852 

2292-222 

618 

019 

2292  593 

2292-903 

2293-333 

2233-704 

2294-074 

2294-444 

2294-815 

2-J9.V1X5 

2295-556 

2295-S-26 

619 

620 

229o-296 

2290-607 

2297-037 

2297-407 

2297-778 

2298-148 

2298-519 

2298  889 

2299-259 

2299630 

620 

G21 

23^0- 

2300-370 

2300-741 

2301-1  n 

2301-481 

2301-852 

2302-222 

2302-693 

2302-963 

2303-833 

621 

623 

23  (3-704 

2304-074 

2304-444 

2304-815 

2305-185 

2305-556 

2305-926 

2306296 

•2306-007 

2307-037 

622 

623 

23)7  -407  12.307  -77  S 

23)8-1  48 

2308-519 

230S-8S9 

•2309-259 

2309  G3( 

2310- 

2310-370 

2310-741 

«V23 

624 

2311  111 

2311-481 

2311-852 

2312-222 

2312-593 

2312-90.3 

2313333 

2313-704 

2314-074 

2314-444 

624 

625 

2314815 

2315-185 

23  1.Y  .-,:,•, 

2315-926 

2316-296 

2316-667 

2317-037 

2317-407 

2317-778 

K318-148 

6-.>5 

G2G 

2318519  2318-8S9 

211  9-253 

2319-630 

2320- 

2320-370 

2320-741 

2321-111 

•2321-481 

2321-852 

626 

c>n 

J322-222 

2322*593 

2322-903 

2323-333 

2323704 

2324-074 

2324-444 

•2324-815 

2325-185 

2325-556 

627 

62  •» 

2325926 

232>;"2'JC 

2320-067 

2327-037 

2327  -4')7 

2327-778 

2328-148 

2328-519 

l'o28-889 

2329-259 

628 

02J 

2329-630 

2330- 

2330-370 

2330741 

2331-111 

•2331481 

2331-852 

2:'32-222 

23i2-593 

2332-963 

629 

030 

2333-333 

2333-70  4 

2334-074 

2334-444 

2334-815 

2335-185 

2335-556 

2335-926 

233629t 

2336-667 

C30 

(531 

2337037 

2337-407 

2337-778 

2338-148 

2338-519 

2338-889 

2339-259 

•2339-f30 

2340- 

2340-370 

P31 

(532 

2340741 

2341-111 

2341-4S1 

3341-852 

2ii  12-22- 

•2342-5fj:; 

2342-9GG 

•2343-G33 

2343-704 

•2344-074 

632 

033 

-1314-414 

2344-815 

2345-185 

2o45-55G 

23LV920 

2346-290 

•234G-GG7 

2347  037 

•2347-407 

•2347-778 

G33 

634 

234vl48 

2348519 

2318-889 

2343-25) 

2319  630 

2350- 

2350-370 

2350741 

•2351-111 

2351-481 

634 

035 

2351-852 

2352-222 

2352-593 

J352-9G3 

2353-33:i 

23'  3  704 

2354-074 

2,?54-444 

2G54815 

2355-185 

635 

G3l> 

2.355-556 

2:555926 

235G-29G 

2356-667 

•2:j.-.7-o.:7 

23;.7-407 

2357-778 

23;58-148 

•2358519 

2358889 

636 

037 

2359-259 

2359-630 

2360- 

2360-370 

2G60-741 

2361-111 

2361-481 

^361-852 

2362-222 

23C2-593 

637 

G!8 

:;i-.2-.»  -3 

2363-333 

2363-704 

2364-074 

2304-444 

2.364-815 

23G5-185 

236555P. 

•23C5-92* 

23fO-296 

638 

G:59 

2366-6OT 

2367-037 

2307-4  V? 

J3G7-778 

2368-148 

2368-519 

2368-8S9 

23C9-2/9 

2c69-(30 

2370- 

639 

010 

2370-370 

2370-741 

2371-111 

2371-481 

2371-852 

2372-222 

2u7  2-693 

2372-963 

2373-333 

2373704 

C40 

641 

2374-074 

2374-444 

2374-815 

2375-185 

2375-556 

2375926 

2376-29G 

2376-f67 

2377-OS7 

2377-407 

641 

G4i 

2377-778 

2378-148 

2378-519 

2378-.S89 

2379-2C9 

2379-030 

£380- 

2380-370 

2350741 

2381-111 

(42 

013 

23SI-481 

2381-852 

23S2-222 

238259:) 

•J382-9G3 

2.J83-333 

•23*3-704 

2384-074 

2384-444 

2384-815 

643 

014 

2385-  1  85 

2385-556 

23S5-92G 

2386-296 

2386-067 

1387  037 

2387-407 

2387-778 

2388-148 

2C88  51» 

644 

Gl.i 

'38S-8S9 

23S92J9 

2389-03:) 

2393- 

2390-370 

2390-741 

2391-111 

2C91-481 

2391-852 

2392-222 

645 

(UG 

2392-693 

2332-963 

2333-333 

2393-704 

2.394-074 

2394-444 

2394-81;' 

2S95-1!S5 

•2395-550 

2£95-9-.'Q 

646 

oir 

J39I5-29G 

239:5-007 

2397-037 

2337-407 

2397*778 

2398-148 

2398-519 

2398-889 

23t»9-259 

2l!-99  630 

647 

(513 

2400- 

2400-370 

2403-741 

2401-111 

2401-481 

2401-852 

2402-222 

2402-593 

2402-963 

2403-333 

648 

o;j 

2403-704 

2404-074 

2404-444 

2404-815 

2405-185 

2405-550 

2405-920 

2406-296 

2406-607 

2407-037 

649 

050 

24U7-407 

2407-778 

2408-148 

2408-519 

2408-889 

2409-259 

2409-630 

2410- 

2410-370 

2410-741 

050 

G51 

2111-111 

2411-481 

2411-852 

2412-222 

2412-593 

2412-903 

2413-333 

2413-704 

2414-074 

2414-444 

651 

152 

2,14815 

2415-185  2115-550 

2415".;2  i 

2410-290 

2416-667 

2417-037 

2417-407 

2H7-778 

2418-148 

6C2 

GC.3 

211S-519 

2418-883  2419  259 

2419-630 

2420- 

2120-370 

2420-741 

2121-111 

2121481 

2J2l-8i2 

653 

(.51 

2122222 

2:22-59:J  2'22-9i'3 

2123-333 

2123704 

2124-074 

2124-444 

2424-815 

2  '2,5-1  85 

2J2555G 

054 

G55 

242V923 

2420-230 

242J-6G7 

2527-037 

2  '27-407 

2427-778 

2428-148 

2428-519 

2428-889 

2429  2-"9 

055 

G'»0 

2121-63) 

2130- 

2430-370 

2  430-741 

2431-111 

2431-481 

2131-852 

24.?2-222 

2432-593 

2432-903 

656 

0'7 

2.133-333 

2433-704 

2LU-H74 

2434  444 

2!34-S15 

2  135-185 

2435-556 

2435-J126 

2430-296 

2436-667 

657 

058 

2137-037 

2437-407  2437-778 

2438-1  4« 

2138-519 

2438-889 

2439-2-9 

2439-630 

2!40- 

2440-370 

658 

659 

2140-741 

2441-m  2441-481 

2441-8^2  24-12  222 

2442593 

2442963 

2443-33.3 

2443-704 

2144-074 

659 

COO 

2444-444 

2444-815  ^445-185 

2445-550 

2445-926 

2446-296 

2446-C67 

2447037 

2447-407 

2447-778 

GGO 

M.A. 

•0 

•1 

•a 

•3 

•i 

•5 

•G 

•7 

•8 

•9 

M.A. 

MEAN  AREAS   6OI  to  660. 

184 


RULES   FOR    THE   MEASUREMENT   OP   EARTHWORKS. 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  1OO  FEET  IN  LENGTH. 


M.A. 

•0 

«1 

•» 

«3 

.* 

•5 

•6 

•I 

•8 

•9 

ALA. 

fiGl 

2448-148 

2448-519 

244S-889 

2449-259 

2449-630 

2450- 

2450-370 

450-741 

2451-111 

2451-481 

601 

662 

2451-852  2452-222 

2452-593 

2452-963 

2453-333 

2453-704 

2454074 

434-444  2454-815 

2455-185 

602 

063 

2455-556  2455-926 

2456-2% 

2456-667 

2457-037 

2457-407 

2457-778 

2458-148 

2458-519 

2458-889 

663 

664 

2459-259  2459-630 

2460- 

2460-370 

2460-741 

2461-111 

2461-481 

461-852 

2462-222 

2462-593 

664 

665 

2462-963  2463-333 

2463-704 

2464-074 

2464-444 

2464-815 

2465-185 

2465-556 

2465-926 

2466-296 

665 

666 

2466  667  2467-037 

2467-407 

2467778 

2468-148 

2468-519 

2468  889 

2469-259 

2469-630 

2470- 

666 

667 

2470-370  2470-741 

471-111 

2471-481 

2471-852 

2472-222 

2472-593 

2472-963 

2473-333 

2473704 

6(17 

668 

2474-074  2474-444 

474-815 

2475-185 

2475-556 

2475-926 

2476-296 

2476-667 

2477037 

2477-407 

668 

669 

2477-778 

2478148 

478-519 

2478-889 

2479-259 

2479-630 

2480- 

2480-370  2480-741 

2481-111 

669 

670 

2481-481 

2481-852 

482-222 

2482593 

2482-963 

2483-333 

2483-704 

2484-074 

2484-444 

2484-815 

670 

671 

2485-185 

2485-556 

485-926 

2486-296 

2486-667 

2487-037 

2487-407 

2487-778 

2488-148 

2488-519 

671 

672 

2488-889 

2489-259 

489-630 

2490- 

2490-370 

2490-741 

2491-111 

2491-481 

2491-852 

2492-222 

672 

673 

2492-593  j  2492-963 

493-333 

2493  704 

249HT74 

2494-444 

2494815 

2495-185 

2495-556 

2495-9-26 

673 

674 

496-296;  2496-667 

497-037 

2497-407 

2497-778 

2498-148 

2498-519 

2498-889 

2499-259 

2499630 

674 

675 

500-   12500-370 

500-741 

2501-111 

2501-481 

2501-852 

2502-222 

2502-593 

2502-963 

2503-333 

675 

676 

503-704 

.504-074 

504-444 

2504-815 

2505-185 

2505-556 

2505-926 

2506-296 

2506-667 

2507-037 

676 

677 

507-407 

507-778 

508-148 

2508-519 

2508-889 

2509-259 

2509-630 

2510- 

2510-370 

2510-741 

677 

678 

511-111 

511-481 

511-852 

2:M2-222 

2512-593 

2512-963 

2513-333 

2513-704 

2514-074 

2514-444 

678 

679 

514-815 

515-185 

515-556 

'515-926 

2516-296 

2516-667 

2517-037 

2517-407 

2517-778 

2518-14* 

679 

630 

518-519 

518-889 

519-259 

2519-030 

2520- 

2520-370 

2520-741 

25-21-111 

2521-481 

2521-852 

680 

6S1 

522-222 

522-593 

522-963 

2523-333 

2523'  704 

2524-074 

2524-444 

2524-815 

2525-185 

2525-556 

681 

632 

2525-926 

526-296 

526-667 

2527-037 

2527-407 

2527-778 

2528-148 

2528-519 

2528-889 

2529-259 

682 

683 

2529-630 

530- 

530-370 

2530-741 

2531-111 

2531-481 

2531-852 

2532-222 

2532-593 

2532-963 

683 

68  1 

533-333 

533-704 

534-074 

2534-444 

2534-815 

2535-185 

2535-556 

2535-926 

2536-296 

2536-667 

684 

685 

2537-037 

537-407 

537-778 

2538-148 

2538-519 

2538-889 

2539-259 

2539-630 

2540- 

2540-370 

685 

686 

2540-741 

541-111 

541-481 

2541-852 

2542-222 

2542-593 

2542-963 

2543-333 

2543-704 

2544-074 

686 

687 

2544-444 

544-815 

545-185 

2545-556 

2545-926 

2546-296 

2546-667 

2547-037 

2547-407 

2547-77* 

687 

688 

2548-148 

548-519 

2548-8S9 

2549-259 

2549630 

2550- 

2550-370 

2550-741 

2551-111 

2551-4S1 

688 

689 

2551-85-2 

552-222 

2552-593 

2552-963 

2553-333 

2553-704 

2554-074 

2664*444 

2554-815 

2555-1  85 

689 

690 

2555-556 

555-926 

2556-296 

2556-667 

2557-037 

2557-407 

2557-778 

2558-148 

•2558-519 

2558-889 

690 

601 

2559-259 

2559-630 

2560' 

2560-370 

2560-741 

2561-111 

2561-481 

2561-852 

2562-222 

2562-593 

691 

692 

2562963 

2563-333 

2563-704 

2564-074 

2564-444 

2564-815 

2565-185 

2565'556 

2565-926 

2566-296 

692 

6rJ3 

2566-667 

2567-037 

2567-407 

2567-778 

2568-148 

2568-519 

2568-889 

2569-259 

2569-630 

2570- 

693 

694 

2570-370 

2570-741 

2571-111 

2571-481 

2571-852 

2572-222 

2572-593 

2572-963 

2573-333 

2573-704 

694 

695 

2574-074 

2574-444 

2574-815 

2575-185 

2575-556 

2575-926 

2576-296 

2576-667 

2577-037 

2577-407 

695 

696 

2577-778 

2578-148 

2578-519 

2578-889 

2579-259 

2579-630 

2580- 

2580-370 

'2580-741 

2581-111 

696 

697 

2581-481 

2581-852 

2582-222 

2582-593 

2582-963 

2583-333 

2583-704 

2584-074 

2584-444 

2584-815 

697 

698 

2585-185 

2585-556 

2585-926 

2586-296 

2586-667 

2587-037 

2587-407 

2587-778 

2588-148 

2588-519 

698 

699 

2588-889 

2589-259 

2589-630 

2590- 

2590-370 

2590-741 

2591-111 

2591-481 

2591-852 

2592-22L 

699 

700 

2592-593 

2592-963 

2593333 

2593-704 

•2591-074 

2594-444 

2594-815 

259^-185 

2595-556 

2595-926 

700 

701 

•2596-296 

2596-667 

2597-037 

2597-407 

2597-778 

2598-148 

2598-519 

2598-889 

•2599-259 

2599-630 

701 

702 

2600- 

2600-370 

2600741 

2001-111 

2601-481 

2601-852 

2602-222 

2602-593 

•2602-963 

2603-333 

702 

703 

2603-704 

2604-074 

2604-444 

2604-815 

2605-185 

2605-556 

2605-926 

•2606-296 

2606-667 

2607-037 

703 

704 

2607-407 

2607-778 

2608-148 

2608-519 

2608-889 

2609-259 

2609-630 

2610- 

2610-370 

2610-741 

704 

705 

2611-111 

2611-481 

2611-852 

2612-222 

2612-593 

2612-963 

2613-333 

2613-704 

2614-074 

•2614-444 

705 

7  .16 

•2614-815 

2615-185 

2615-556 

2615-926 

2616-296 

2616-667 

2617-037 

2617-407 

2617-778 

2618-148 

706 

707 

2618-519 

2618-889 

2619-259 

2619-630 

2620- 

2620-370 

2620-741 

2621-111 

2621-481 

2621-852 

707 

708 

2622-222 

2622-591, 

2622-963 

2623-33? 

•26-23-704 

2624-074 

2624-444 

2624-815 

2625-185 

2625-556 

708 

709 

2625-920 

2626-296 

2626-667 

26-27-037 

2627-407 

•2627-778 

2628-148 

2628-519 

2628-889 

2629-259 

709 

710 

2629-630 

2630- 

2630-370 

2630-741 

2631-111 

2631-481 

2631-852 

2632-222 

2632-593 

2632-963 

710 

711 

2633-33 

2633-70 

2634-07-1 

2634-444 

2634-815 

2635-185 

2635-556 

2635-926 

2636-296 

2636-667 

711 

712 

•_:637-G3 

2637-40 

2637-77S 

•2638-148 

2638-519 

2038-889 

2639-259 

2639-630 

2640- 

2640-370 

712 

713 

2640-74 

2641-111 

2641-481 

2641-852 

2642-222 

2642-593 

2642-963 

2643-333 

2643-704 

2644-074 

713 

714 

2614-444 

2644-Slo 

2645-185 

2645-556 

2645-926 

2646-296 

2646-667 

2647-037 

2647-407 

2647-778 

714 

715 

2648-14 

2648-519 

2648-880 

2649-259 

2649-630 

2650- 

2650-370 

2650-741 

2651-111 

2651-481 

715 

716 

2651-85 

2652-22' 

2652-593 

2652-963 

2653-333 

2653-704 

2654-074 

2654-444 

2654-815 

2655-185 

716 

717 

2655-55 

2655-926 

2656-29(3 

2656-667 

2657-037 

2657-407 

2657-778 

2658-148 

2658-519 

2658-889 

717 

718 

2659-25 

2659-630 

•2660- 

2660-370 

2660-741 

2661-111 

2661-481 

2661-852 

2662-222 

2662-593 

718 

719 

2662-96 

2363-333 

2663-704 

2664-074 

2664-444 

2664-815 

2665-185 

2665-556  2665-926 

2666-296 

719 

720 

2C66-66 

2667-03" 

2667-407 

2607-778 

2068-148 

2668-519 

2668-889 

2669-259 

2669-630 

2670- 

720 

M.A. 

•0 

•1 

•0 

•3 

•4 

•5 

•  »6 

•7 

•8 

•9 

M.A. 

MEAN  AREAS  661  to  7XO. 

RULES   FOR   THE   MEASUREMENT    OF    EARTHWORKS. 


J85 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  1OO  FEET  IN  LENGTH. 


.M.A. 

•0 

•  1 

•a 

•3 

•4: 

•5 

•6 

•7 

•8 

•9 

M.A. 

7-21 

2070-370  2670-741 

2671-111 

2671-481 

2671-852 

2tf7  2-222 

2672-593 

2672-963 

2673-333 

2673-704 

72I 

72-2 

2074-074  2674-444 

2674-815 

2675-185 

2675-556  12675-926 

2676-296 

2670-607 

2677-037 

2677-407 

722 

723 

2G77-778  2678-148 

2678-519 

2678889 

2679-259 

2079-630 

2680- 

2680-370 

2680-741 

2681-111 

723 

724 

26S1-481  2681-852 

2682-222 

2082-59:5 

2682-963 

2683-333 

2683-704 

2684-074 

2684-444 

2684-815 

724 

725 

2685-185  2685-556 

2685-926 

2680-296 

2686-667 

2687-037 

2687-407 

2687-778 

26-8-148 

2688-519 

725 

720 

2088-889  2689-259 

2689-630 

2690- 

2690-370 

2690-741 

2691-111 

2691-481 

2691-852 

2092-222 

726 

727 

2(192-593  2692  963 

2693-333 

2693-704 

2694-074 

2694-444 

2694-815 

2695-185 

2695-556 

2695-926 

727 

728 

2096-296  2696  007 

2697-037 

2697-407 

2697-778 

2698-148 

2698-519 

2698-889 

2699-259 

2699-630 

728 

729 

2700- 

2700-370 

2700-741 

2701-111 

2701-481 

2701-852 

2702-222 

2702-593 

2702-963 

2703333 

7-29 

730 

2703-704 

2704-074 

2704-444 

2704-815 

2705-185 

2705-556 

2705-926 

2706-296 

2706-667 

2707-037 

730 

731 

2707-407 

2707-778 

2708-148 

2708-519 

2708-889 

2709-259 

2709-630 

2710- 

2710-370 

2710-741 

731 

732 

2711-111  J2711-481 

2711-852 

2712-2-22 

2712-593 

2712-963 

2713-333 

2713-704 

2714-074 

2714-444 

732 

733 

2714-815  '27  15-  185 

2715-550 

2715-926 

2716-296 

2716-667 

2717-037 

2717-407 

2717-778 

2718-148 

733 

7*4 

2718-519  2718-889 

2719-259 

2719-630 

2720- 

2720-370 

2720-741 

2721-111 

2721-481 

2721-852 

734 

735 

2722  222  2722-593 

•2722903 

272:5-:!:;;; 

2723-704  127  24-H74 

2724-444 

2724815 

2725-185 

2725-556 

735 

736 

2725-926  2726-296 

2726-607 

2727-037 

2727-407 

2727-778 

2728-148 

2728-519 

2728-889 

2729-259 

736 

737 

2729030;  27  30- 

2730-370 

2730-741 

2731-111 

2731-481 

2731-852 

27:',2-2-22 

2732-593 

2732-963 

737 

738 

27:33-333  2733-704 

2734-1  '74 

2734-444 

2734-815 

2735-185 

2735-556 

2735-926 

2736-296 

2736-667 

738 

739 

2737-037  27:  17-407 

2737-778 

2738-148 

2738-519 

2738-889 

2739-259 

2739-630 

2740- 

2740-370 

739 

740 

2740-741 

2741-111 

2741-481 

2741-852 

2742-222 

2742-593 

2742-963 

2743-333 

2743-704 

2744-074 

740 

741 

2744-444 

2744-815 

2745-185 

2745-556 

2745-926 

2746-296 

2746-667 

2747-037 

2747-407 

2747-778 

741 

742 

2748-148  2748-519 

2748-889 

2749-259 

2749-630 

2750- 

2750370 

2750741 

2751-111 

2751-481 

742 

743 

2751-852  2752-222 

•27.V2-f>!KJ 

2752-963 

2753-333 

2753-704 

2754-074 

2754-444 

2754-815 

2755-185 

743 

744 

2755-556  2755-926 

2756-296 

2756-667 

2757-037 

2757-407 

2757-778 

2758  148 

2758-519 

2768-888 

744 

745 

2759-259  2759-630 

2760- 

2760-370 

2760741 

2761-111 

2761-481 

2761-852 

2762-222 

2762-593 

745 

746 

2762-963  2763-333 

27K3-704 

2764-074 

27  64-444  12764-815 

2765-185 

2765  550 

2765-926 

2766-296 

746 

747 

2766-667  2767'037 

2767-407 

2767-778 

2768-148|2768-51l) 

2768-889 

2769-259 

2769-630 

2770- 

747 

748 

2770-370  2770-741 

2771-111 

2771-481 

2771-852 

2772-222 

2772-593 

2772-963 

2773-333 

2773-704 

748 

749 

2774-074  2774  444 

2774-815 

2775-185 

2775-556 

2775-926 

2776-296 

2776-667 

2777037 

2777-407 

749 

750 

2777-778 

2778-148 

2778-519 

2778-889 

2779-259 

2779-630 

2780- 

2780-370 

2780-741 

2781-111 

750 

751 

2781-481 

2781-852 

2782-222 

2782-593 

2782-963 

2783-333 

2783-704 

2784-074 

2784-444 

2784-815 

751 

752 

2785-185 

2785-556 

2785-926 

2786-296 

2786-667 

2787-037 

2787-407 

2787-778 

2788-148 

2788-519 

752 

753 

2788-889 

2789-259 

2789-030 

2790- 

2790-370 

2790-741 

2791-111 

2791-481 

2791-852 

2792-222 

753 

754 

2792-593 

2792-963 

2793-333 

2793-704 

2794-074 

2794-444 

2794-815 

2795-185 

2795-556 

2795-926 

754 

755 

2796-296 

2796-667 

2797-037 

2797-407 

2797-778 

2798-148 

2798-519 

2798  889 

2799-259 

2799-630 

755 

756 

2800- 

2800-370 

2800-741 

2801-111 

2801-481 

2801-852 

2802-222 

2802-593 

2802-963 

2803-333 

756 

757 

2803-704 

2804-074 

2S04-444 

2804-815 

2805-185 

2805-556 

2805-926 

2806-296 

2806-667 

2807037 

757 

758 

2807-407 

2807-778 

2808-148 

2808-519 

2808-889 

2809-259 

2809-630 

2810- 

2810-370 

2810-741 

758 

759 

2811-111 

2811-481 

2811-852 

2812-222 

2812-593 

2812-963 

2813-333 

2813-704 

2814-074 

2814-444 

759 

760 

2814-815 

2815-185 

2815-556 

2815-926 

2816-296 

2816-667 

2817-037 

2817-407 

2817-778 

2818-148 

760 

761 

2818-519 

2818-889 

2819-259 

2819-630 

2820- 

2820-370 

2820-741 

282M11 

2821-481 

2821-852 

761 

762 

2822-222  2822-593 

2822-963 

2823-333 

2823-704 

2824-074 

2824-444 

2824-815 

2825-185 

2825-556 

762 

763 

2825-926  2826-296 

2826-667 

2827-037 

2827-407 

2827-778 

2828-148 

2828-519 

2828-889 

2829-259 

763 

764 

2829-630  2830- 

2830-370 

2830-741 

2831-111 

2831-481 

2831-852 

2832-222 

2832-593 

2832-963 

764 

765 

2833-33312833-704 

2834-074 

2834-444 

2834-815 

2835-185 

2835-556 

2835-926 

2836  296 

2836  667 

765 

766 

2837-037 

2837-407 

2837-778 

2838-148 

2838-519 

2838-889 

2839-259 

2839-630 

2840- 

2840-370 

766 

767 

2840-741 

2841-111 

2841-481 

2841-852 

2842-222 

2842-593 

2842-963 

2843-333 

2843-704 

2844-074 

767 

7G8 

2844-444 

2844-815 

2845-185 

2845-.r>56 

2845-920 

2846-296 

2846667 

2847-037 

2847-407 

2847-778 

768 

769 

2848-148 

2848-51912848-889 

2849-259 

2849-630 

2850- 

2850-370 

2850-741 

2851111 

2851-481 

769 

770 

2851-852 

2852-222 

2852-593 

2852-963 

2853-333 

2863-704 

2854-074 

2854-444 

2854-815 

2855-185 

770 

771 

2855-556 

2855-926 

2856-296 

2856-667 

2857-037 

2857-407 

2857-778 

2858-148 

2858-519 

2858-889 

771 

772 

2859-259 

2859-630 

2860- 

2860-370 

2800-741 

2861-111 

2861-481 

2861-852 

2862-222 

2802-593 

772 

773 

2862-963 

2863-333 

2863-704 

2864-074 

2864-444 

2864-815 

2865-185 

2865-556 

2865-926 

2866-296 

773 

774 

2866-667 

2867-037 

2867-407 

2807-778 

2868-148 

2868-519 

•>t;s-s,v, 

2869-259 

2869-630 

2870-  ' 

774 

775 

2870-370 

2870-741 

2871-111 

2871-481 

2871-852 

2872-222 

2872-593 

2872  963 

2873-333 

2873-704 

775 

776 

2874-074 

2874-444 

2874-815 

2875-185 

2875-556 

2875-926 

2876-296 

2876-667 

2877-037 

2877-407 

776 

777 

2877-778 

2878-148 

2878-519 

2878-889 

2879-259 

2879-630 

2880- 

2880-370 

2880-741 

2881-111 

777 

778 

2881-481 

2881-852 

2882-222 

28S2-593 

2882-963 

2883-33312883-704 

2884-074 

2884-444 

2884-815 

778 

779 

2885-185 

2885-556|2885-926 

2886-296 

2886-667 

•js»7-n:;7  2»87-407 

2887-778 

2888-148 

2888-519 

779 

780 

2888-889 

2889-259  2889-630 

2890- 

2890-370 

2890-741 

2891-111 

2891-481 

2891-852 

2892-222 

780 

M.A 

•0 

•1 

•» 

•3 

•*. 

•5 

•6 

•7 

•8 

•9 

M.A. 

MEAN  AREAS  721  to  78O. 

18G 


RULES   FOR   THE   MEASUREMENT   OP   EARTHWORKS. 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  WO  FEET  IN  LENGTH. 


M.A. 

•0 

•1 

•a 

•3 

«4 

•5 

•6 

•7 

•8 

•9 

M.A. 

781 

2892-593 

2892-963 

893-333 

2893-704 

2894-074 

2894-444 

.'8S4-815 

895-185 

2895-556 

^895-92*', 

781 

782 

2S96-296  2S96  667 

2897-037 

2897-407 

2897-778 

2898148 

2898-519 

2898-889 

2899-259 

2899-630 

782 

783 

2900-    2900-370 

2900-741 

2901-111 

2901-481 

2901  852 

2902-222 

2902-593 

2902-963 

2903-3:>3 

res 

784 

2903-704  -2904-074 

904-444 

2904-815 

2905-185 

2905-556 

-905-926 

2906-296 

2906-667 

2907-037 

784 

785 

2907-407  2907-778 

908-148 

2908'5I9 

2908-889 

2909-259 

2909-630 

2910- 

2910-370 

2910-741 

785 

786 

2911-111  2911-481 

911-852 

2912-222 

2912-593 

•2912-963 

2913-333 

2913-704 

2914-074 

2914-444 

7^6 

787 

2914-815  2915  185 

915-556 

2915-926 

2916-296 

2916-667 

2917-037 

2917-407  2917-778 

2918-148 

787 

788 

2918-519  -2918-889 

919-259 

2919-630 

2920- 

2920-370 

2920741 

2921-111  1  2921-481 

2921  -852 

788 

789 

2922-222  12922-593 

>922-963 

29-'3-333 

2923-704 

2924-074 

-1924-444 

2924-815 

2925  185 

2925-55G 

789 

790 

2925-926 

2926-296 

2926-667 

2927-037 

2927-407 

2927-778 

2928-148 

2928-519 

2928-889 

2929-259 

790 

791 

2929-630 

2930- 

2930-370 

2930-741 

2931-111 

2931-481 

2931-852 

2932-222 

2932-593 

2932-963 

791 

792 

2933-333  2933-704 

2934-074 

2934-444 

2934-815 

2935-185 

2935-556 

2935-926 

2936-296 

2936-667 

792 

793 

2937-0:37  2937-407 

2937-778 

2938-148 

2938-519 

2938-889 

2939-259 

2939  630 

2940- 

2940-370 

793 

704 

2940-741  2941-111 

2941-481 

2941-852 

2942-222 

2942-593 

2942-963 

2943-333 

2943-704 

2944-074 

7^4 

795 

2944-444  1'29  44-815 

2945-185 

_'945-556 

2945-926 

J946-296 

2946-667 

2947-037 

2947-407 

2947-778 

795 

796 

2948-148 

2948-519 

2948-889 

2949-259 

2949-630 

2950- 

2950-M70 

2950-741 

2951-111 

2951-481 

79fi 

797 

2951-852 

29.V2-222 

2952-593 

2952-963 

2953-333 

2953704 

2954-074 

^954-444 

2954-815 

2955185 

797 

798 

2955-556 

2955-926 

2956-296 

2956-667 

2957-037 

2957-407 

2957-778 

2958-148 

2958-519 

2958-889 

798 

799 

2959259 

2959-630 

2960- 

2960-370 

2960-741 

2961-111 

2961-481 

2961-852 

2962-222 

2962-593 

799 

800 

2962-963 

2963-333 

2963-704 

2964-074 

2964-444 

2964-815 

2965-185 

2965-556 

2965-926 

^966-296 

800 

801 

2966667 

2967-037 

2967-407 

2967-778 

2968-148 

2968-519 

2968-889 

2969-259 

2969-630 

2970- 

801 

802 

2970-370 

2970-741 

2971-111 

2971-4S1 

2971-852 

2972-222 

2972-593 

2972963 

2973-033 

2973-704 

802 

803 

2974-074 

2974441 

2974-815 

2975-185 

2975-556 

2975-926 

2976-296 

2976667 

•2977-037 

2977-407 

803 

801 

2977-778 

2978-148 

2978-519 

2978-889 

2979-259 

2979T30 

2980- 

2980-370 

2980-741 

2981-111 

804 

805 

2981-481 

2981-852 

2982  222 

2982-593 

2982-963 

2983-333 

2983-704 

2984-074  1  2984-444 

2984-815 

805 

806 

2985-185 

2985556 

2985-926 

2986-296 

29S6-667 

2987-037 

2987-407 

2987-778 

2988-148 

2988-519 

806 

807 

2988-889 

29H9-259 

2989-630 

2990- 

2990  370 

2990-741 

2991-111 

2991  -4S1 

2991-85? 

2992-222 

807 

808 

2992593 

2992-963 

2993-333 

2993-704 

2994-074 

J994-444 

2994-815 

2995-185 

2995-556 

2995-921 

808 

809 

2996-296 

2996-667 

2997-037 

2997-407 

2997-778 

2998-148 

2998-519 

2998-889 

2999-259 

2999-031 

809 

810 

3000- 

3000-370 

3000-741 

3001-111 

3001-481 

3001-862 

3002-222 

3002-593 

3002-963 

3003-333 

810 

811 

3003-704 

3004-074 

3004-444 

3004-815 

3005-185 

3005-5.56 

3005-926 

3006-296 

3006-667 

3007-037 

811 

812 

3007-407 

3007-778 

3008-148 

3008519 

3008-889 

3009-259 

3009'63( 

3010- 

3010-370 

3010-741 

812 

813  |3011-111 

3011-481 

3011-852 

3012-222 

3012-593 

301  2-963 

301333: 

3013-704 

3014-074 

3014-444 

813 

814 

3014-815 

.•5015-185 

3015556 

3015-926 

3016-296 

3016-667 

3017-037 

3017407 

3017-778 

3018-148 

814 

815 

3018-519 

3018-889 

3019-259 

3019-630 

3020- 

3020-370 

3020-741 

3021-111 

3021-481 

3021-852 

815 

816 

302-2-222 

3022-59} 

3022-963 

3023-333 

3023-704 

3024-074 

3024444 

3024-815 

3025-185 

3025-556 

816 

817 

3025-926 

3026-296 

3026-667 

3027-0.57 

3027-407 

3027-778 

3028148 

3028-519 

3028-889 

3029-259 

817 

818 

3029-630 

3030- 

3030-370 

3030-741 

3031-111 

3031-481 

3031-852 

3032-222 

3032-593 

3032-963 

818 

819 

3033-333 

3i  133-7  04 

3034-074 

3034-444 

3034-815 

3035-185 

3035-556 

3035-926 

3036-296 

3036-667 

819 

820 

3037-037 

3037-407 

3037-778 

3038-148 

3038-519 

3038-8SP 

3039-259 

3039  630 

3040- 

3040-370 

820 

821 

3040-741 

3041-111 

3041-481 

3041-852 

3042-222 

.1042-593 

3042-963 

3043-333 

3043-704 

3044-074 

821 

822 

3044-444 

3044-815 

3045-1  85 

3045-551 

3045-926 

3046-296 

3046-667 

3047-037 

3047-407 

3047-  77  S 

822 

823 

3048-148 

3048-519 

3048-889 

3049259 

3049630 

3050- 

3050370 

3050-741 

3051-111 

3051-481 

823 

824 

3051-852 

305222- 

3052-693 

3052-963 

3053-333 

3053-704 

3054-074 

3054-444 

3054-815 

3055-185 

824 

825 

3055-556 

3055-926 

3056296 

3056-067 

3057-037 

3057-407 

3057-778 

3058-148 

3058-519 

3058-889 

825 

826 

3059259 

3059-630 

30(30- 

3060-370 

3060-741 

3061  -111 

3061-481 

3061-852 

3062-222 

3062-593 

826 

827 

3062-963 

3063-33; 

3063-704 

3064-074 

3064-444 

3064-815 

3065-185 

3065-556 

3065-926 

3066296 

827 

828 

3066-667 

3067-037 

3067-407 

3067-778 

3068-148 

3068-519 

3068-889 

3069259 

3069-630 

3070- 

828 

829 

3070-371 

3070-74 

3071-111 

3071-481 

3071-852 

3072-222 

3072-593 

3072-963 

3073-333 

3073-704 

829 

830 

3U74-074 

3074-444 

3074-815 

3075-185 

3075-556 

3075-926 

3076-296 

3076-667 

3077-037 

3077-407 

sao 

R31 

?077-778 

3078-148 

3078-519 

3078-889 

3079-259 

3079-630 

3080- 

3080-370 

3080-741 

308M11 

831 

832 

3081-481 

3081-85- 

3082-222 

3082-593 

3082-9*3 

3083-333 

3083-704 

3084-074 

3084-444 

3084-815 

832 

833 

3085-1  ?5 

3085-556 

3085-926 

•W6-296 

3086-667 

3087-037 

3087-407 

3087-778 

3088148 

3088-519 

833 

834 

3088-889 

3089-259 

3089-630 

3090- 

3090-370 

3090-741 

3091-111 

3091-481 

3091-85-2 

3092-222 

834 

8i!5 

3092-593 

3092-96: 

3093-333 

3093-704 

3094-074 

3094-444 

3<i94-81f 

3095-185 

3095-556 

3095-926 

835 

836 

3096-296 

3096-66- 

30,97-037 

3097-407 

3097-778 

3098-148 

3098-519 

3098  889 

3099-259 

3099-630 

836 

837 

3100- 

3100-370 

3100-741 

3101-111 

3101-481 

3101-852 

3102-222 

31  02-593  13102-963 

:-,103333 

837 

838 

3103-704 

3104-074 

3104-444 

3104-815 

3105-185 

3105-556 

3105-926 

3106-296 

3106-667 

3107-037 

838 

839 

3107-407 

3107-778 

310S-14S 

3108-519 

3108-889 

3109-259 

3109-630 

3110- 

3110-370 

3110-741 

839 

840 

311  Mil 

3111-481 

3111-852 

3112-222 

3112-593 

3112-963 

3113-333 

3113-704 

3114-074 

3114-444 

840 

M.A. 

•O 

•1 

•a 

•3 

•* 

•5 

•6 

•7 

•  8 

•9 

M.A. 

MEAN  AREAS  781  to  84O. 

RULES   FOR   THE   MEASUREMENT    OF   EARTHWORKS. 


187 


CTTJilC  YARDS  TO  MEAN  AREAS  FOR  WO  FEET  IN  LENGTH. 


M.A- 

•0 

•1 

•a 

•3 

•4 

•5 

•6 

•7 

•  8 

•9 

M.A. 

841 

3114-815 

3115-185 

3115-556 

3115-926 

3116-296 

3116-667 

3117037 

3117-407 

3117-778 

3118-148 

841 

812 

3118  519  3118-889 

3119-259 

3119-630 

3120- 

3120-370 

3120-741 

3121-111 

3121-481 

3121-852 

842 

843 

3122-222  3122  593 

3122-9U3 

3123-333 

3123-704 

3124-H74 

3124-444 

3124-815 

3125-185 

3125-556 

843 

814 

3125-926 

3126-296 

3126-667 

3127-037 

3127-407 

3127-778 

3128148 

3128-519 

3128-889 

3129-259 

844 

845 

3129-630 

3130- 

3130-370 

3130-741 

3131-111 

3131-481 

3131-852 

3132-222 

3132-593 

3132963 

845 

846 

3133-333 

3133-704 

5134-074 

3134-444 

3134-815 

3135-185 

3135-556 

3135-926 

3136296 

3136-667 

846 

847 

3137-037 

3137-407 

3137-778 

3138-148 

3138-519 

3138-889 

3139-259 

3139-630 

3140- 

3140-370 

847 

848 

3140741 

3141-111 

3141-481 

3141-852 

3142-222 

3142-593 

3142-963 

3143-333 

3143-704 

3144074 

848 

849 

5144-444 

3144-815 

3145-185 

3145-556 

3145-926 

3146-296 

3146-667 

3147-037 

3147-407 

3147-778 

849 

850 

3148-  148 

3148-519 

3148-889 

3149-259 

3149630 

3150- 

3150-370 

3150-741 

3151-111 

3151-481 

850 

851 

3151-852 

3152-222 

3152-593 

3152-963 

3153-333 

3153-704 

3154-074 

3154-444 

3154-815 

3155-185 

851 

852 
853 

3  155-556  j  3155-926 
3159-259  3159-630 

3156-296 
3160- 

3156-667 
3160-370 

3157-037 
3160-741 

3157-407 
3161-111 

•3157-778 
3161-481 

3158-148 
3161-852 

3158-519 
3162-222 

3158-889, 
3162-593 

852 
853 

854 

J  102-963  ;  31  63-333 

3163-704 

3164-074 

3164-444 

31(34-815 

3165185 

3165-556 

3165-926 

3166-296 

854 

855 

5166667 

3167-037 

3167-407 

3167-778 

3168-148 

3168-519 

3168-889 

3169-259 

3169-630 

3170- 

855 

856 

3170-370 

3170-741 

3171-111 

3171-481 

3171-852 

3172-222 

3172-593 

3172-963 

3173-233 

3173-704 

1-56 

857 

3174-074 

3174444 

3174-815 

3175-185 

3175-556 

3175-926 

3176-296 

3176-667 

3177-037  13177-407 

857 

858 

3177-778 

3178-148 

3178-519 

3178-889 

Jl  79-259 

3179630 

3180- 

3180-370 

3180741 

3181111 

858 

859 

5181481 

3181-852 

3182-222 

3182-593 

3182-963 

3183-333 

3183-704 

3184-074 

3184-444 

31K4-815 

859 

860 

3185-185 

3185-556 

3185-926 

3186-296 

3186-667 

3187-037 

3187-407 

3187-778 

3188-148 

3188-519 

860 

861 

1188-889 

3189-259 

3189-630 

3190- 

3190-370 

3190-741 

3191-111 

3191-481 

3191-852 

3192-222 

861 

862 

5192-593 

3192-963 

3193333 

3193-704 

3194-074 

3194-444 

3194815 

3195-185 

3195-556 

3195-926 

862 

863 

3196-296 

3196-667 

3197-037 

3197-407 

3197-778 

3198-148 

3198-519 

3198-889 

3199-259 

3199-630 

863 

864 

3200- 

3200-370 

3200-741 

3201-111 

3201-481 

3201-852 

3202-222 

3202-593 

32U2-963 

3203-333 

864 

865 

3203-704 

3204-074 

3204-444 

3204-815 

3205185 

3205-556 

3205-926 

3206-296 

3206-667 

3207-037 

865 

8G6 

'.207-407 

3207-778 

3208-148 

3208-519 

3208-889 

3209-259 

3209-630 

3210- 

3210-370 

3210741 

866 

867 

3211-111 

3211-481 

3211852 

:;2i2"2-2-2 

3212-593 

3212-963 

3213-333 

3213-704 

3214-074 

3214-444 

867 

868 

3214-815 

3215-185 

3215-556 

3215-926 

3216-296 

3216-667 

3217-037 

3217-407 

3217-778 

3218-148 

868 

869 

3218-519 

3218889 

3219-259 

3219-630 

3220- 

3220-370 

3220-741 

3221-111 

3221-481 

3221-852 

869 

870 

3222-222 

3222593 

3222-963 

3223-333 

3223-704 

3224-074 

3224-444 

3224-815 

3225-185 

3225-656 

870 

871 

3225-926 

3226-296 

3226-667 

3227-037 

3227-407 

3227-778 

3228-148 

3228-519 

3228-889 

3229-259 

871 

872 

'3229-630 

3230- 

J230-370 

3230-741 

3-2IU-111 

3231-481 

3231-852 

3232-222 

3232-593 

3232-963 

872 

873 

3233-333 

3233-704 

3234-074 

3234-444 

3234-815 

3235-185 

32&V556 

3235-926 

3236-296 

3236-667 

873 

874 

3237-037 

3237-407 

3237-778 

3238-148 

3238-519 

3238-889 

3239-259 

3239-630 

3240- 

3240  370 

874 

875 

3240  741 

3241111 

3241-481 

3241-852 

3242-222 

3242-593 

3242-963 

3243-333 

3243-704 

3244-074 

875 

876 

52U-444 

3244-K15 

3245-18.") 

3245-550 

3245-926 

3246-296 

3246-667 

3247-037 

3247-407 

3247-778 

876 

877 

3248-148 

3248-519 

3248-889 

3249-259 

3249-630 

3250- 

3250-370 

3250-741 

3251-111 

3251-481 

877 

878 

3251-852 

3-252-222 

3252-593 

3252-963 

3253-333 

3253704 

3254-074 

3254-444 

3254-815 

3255-185 

878 

879 

3255-556 

3255-926 

3256-296 

3256667 

3257-037 

3257-407 

3257-778 

3258-148 

3258-519 

3258-889 

879 

MO 

3259-259 

3259-630 

3260- 

3260-370 

3260741 

3261-111 

3261-481 

3261-852 

3262-222 

3262-593 

880 

881 

3262-963 

3263-333 

3263-704 

3264-074 

3264-444 

3264-815 

3265-1  80 

3265-556 

3265926 

3260-296 

881 

882 
883 

3266-667 
3270-370 

3267-037 
3270-741 

3267-407 
3271-111 

3267-778 
3271-481 

32«8-148 
3271-852 

3268-519 
3272  22-2 

3268-889  3269'259 
3272-59313272-963 

32*9-630 
3273-333 

3270- 
3273-704 

882 
8S3 

884 

3274-074 

3274-444 

3274-815 

3275-185 

3275-556 

3275-9|6 

3276-296 

3276-667 

3277H37 

3277-407 

884 

885 

3277-778  13278-148 

3278-519 

3278-889 

3279-259 

3279-flO 

3280- 

3280-370 

3280-741 

3281-111 

885 

886 

3281-481  3281-852 

3282-222 

3282-593 

3282-963 

3283-333 

3283-704 

3284-074 

3284-444 

3284815 

886 

8S7 

3285-185  32^5-556 

3285-926 

3286-296 

32866R7 

3287-037 

3287-407 

32S7-778 

3288-148 

3288-519 

887 

888 

3288-88913289-259 

32^9-630 

3290- 

3-290-370 

3290-741 

3291-111 

3291-481  3291-852 

3292222 

888 

889 

32<)2-593  3292-9<'3  3293  333 

3293-704 

3294-074 

3294  444 

3294-815 

3295-1  85 

3295-556 

3^95-926 

889 

890 

3296-29t> 

3296-667 

3297-037 

3297-407 

3297-778 

3298-148 

3298-519 

3298-889 

3299-259 

3299-630 

890 

891 

3300- 

3300-370 

3300-741 

3301-111 

3301-481 

3301-852 

3302-222 

3302-593 

3302-963 

3303333 

891 

892 

3303704 

3304-074 

315  14  444 

3304-815 

3315-185 

3305-556 

3305-926 

.'3306-296 

3306-667 

3307-037 

892 

893 

3307-407 

3307-778 

3308-148 

3308-519 

3308-889 

3309-259 

3309  630 

3310- 

3310370 

13310741 

893 

894 

3311*111 

153  11-481 

3311-852 

3312-222 

3312593 

3312-963 

3313333 

3313-704 

13314-074 

3314-444 

894 

895 

31314815 

3315-185 

3315-556 

3315926 

3316-296 

331  6-667 

3317  037 

3317-407 

3317-778 

3318-148 

895 

896 

3318-519 

3318-889 

3319-259 

3319-630 

3320- 

3320-370 

3320-741 

3321-111 

3321-481 

3321-852 

896 

897 

3322-222 

3322-593 

33-22-9(i3 

33-23333 

3323  704 

3324-074 

33-24-444 

3324-S15 

3325-185 

3325-556 

897 

893 
899 

3325-9-2C, 
3329-630 

3326-296 
3330- 

3326-6R7 
3330370 

3327-037 
333V741 

3327-407 
3331-111 

3327-778 
3331  481 

3328-148 
3331-852 

3328-519 
3332222 

3328-889 
3332-693 

3329-259 
3332-963 

898 
899 

900 

3333-333 

3333-704 

=3334074 

3334-444 

3334-815 

3335-185 

3335-556 

3335-926 

3336-296 

3336-667 

900 

M.A 

•0 

•1 

•a 

•3 

•4 

•5 

.6 

•7 

•8 

•0 

M.A. 

MEAN  AREAS   841  to  000. 

188 


RULES   FOR    THE    MEASUREMENT    OF    EARTHWORKS. 


CUBIC  YARDS  TO  MEAN  AREAS  FOR  10O  FEET  IN  LENGTH. 


M.A. 

•0 

•1 

•3 

•3 

•4t 

•5 

•6 

•7 

•8 

•9 

M.A. 

901 

3337-037 

3337-407 

3337-778 

3338-148 

3338-519 

3338-889 

3339-259 

3339-630 

3340- 

3340-370 

901 

902 

3340-741 

3341-  }\\ 

3341-481 

3341-852 

3342-22-2 

3342-593 

3342-963 

3343-333  3343'704 

3344-074 

902 

903 

3344-414 

3344-815 

3345-185 

3345-556 

3345-926 

3346-296 

3346-667 

3347-037  3347-407 

3347-778 

903 

904 

1348-148 

3348-519 

o348-889 

3349-259 

3349-630 

3350- 

3350-370 

3350-741 

3251-111 

3351-481 

904 

905 

3351-852 

3352-222 

3352-593 

3352-963 

3353-333 

"3353-704 

3354-074 

3354-444 

3354-815 

rf355-185 

905 

906 

3355-556 

3355-926 

3356-296 

3356-667 

3357-037 

3357-407 

3357-778 

3358-148  13358-519 

3358-889 

906, 

907 

3359-259 

3359-630 

3360- 

3360-370 

3360-741 

3361-111 

3361-481 

3361-852  ;  3362-222 

3362-593 

907 

908 

3362-963 

3363-333 

3363-704 

3364  074 

3364-444 

3364-815 

3365-185 

33  65-556  '3:365-926 

3366-296 

908 

909 

3366-667 

3367  037 

3367*407 

3367-778 

3368-148 

3368-519 

3368-889 

3369-259;  3369-630 

3370- 

909 

910 

3370-370 

3370-741 

3371-111 

3371-481 

3371-852 

3372-222 

3372-593 

3372963 

3373-333 

3373-704 

910 

911 

3374-074 

3374-444 

3374-815 

3375-185 

3375-556 

3375-926 

3376-296 

3376-667 

3377-037 

3377-407 

911 

912 

3377-778 

3378-148 

3378-519 

3378-889 

3379-259 

3379-630 

3380- 

3380-370  3380-741 

3381-111 

912 

913 

3381-481 

3381-852 

•3382-222 

3382-593 

3382-963 

3383-333 

3383-704 

3384-074  3384-444 

3384-815 

913 

914 

.'33S5-185 

3385-556 

3385-926 

3386-296 

3386667 

3387-037 

3387-407 

3387-778  3388-148 

3388-519 

914 

915 

3388-889 

3389-259 

3389-630 

3390- 

3390-370 

3390-741 

3391-111 

3391-481  3391-852 

3392-222 

915 

916 

3392-593 

3392-963 

3393-333 

3393-704 

3394-074 

3394-444 

3394-815 

3395-185;  3395-556 

3395-926 

916 

917 

3396-296 

3390-667 

3397-037 

3397-407 

3397-778 

3398-148 

3398519 

3398-889!  3399-259 

3399-630 

917 

918 

3400- 

3400-370 

3400741 

3401-111 

5401-481 

3401-852 

3402-222 

3402-593 

3402-963 

3403-333 

918 

919 

3403-704 

3404-074 

3404-444 

3404-815 

3405-185 

3405-556 

3405-926 

3406-296 

3406-667 

3407-037 

919 

920 

3407-407 

3407-778 

3408-148 

3408-519 

3408-889 

3409-259 

3409-630 

3410- 

3410-370 

3410-741 

920 

921 

3411111 

3411-481 

3411-852 

3412-222 

3412-593 

3412-963 

3413-333 

3413-704 

3414-074 

3414-444 

921 

922 

3414-815 

3415-185 

3415-556 

3415-926 

3416-296 

3416-667 

3417037 

3417-407 

3417-778 

3418-148 

922 

923 

3418-519 

3418-889 

3419-259 

3419-630 

3420- 

3420-370 

3420-741 

3421-111 

3421-481 

3421-852 

923 

924 

3422-222 

3422-593 

3422-963 

3423-333 

3423-704 

3424-074 

3424-444 

3424-815 

3425-185 

3425-556 

924 

925 

3425-926 

3426-296 

3426-667 

3427-037 

3427-407 

3427-778 

3428-148 

3428-519 

3428-889 

3429259 

925 

920 

3429-630 

3430- 

3430-370 

3430-741 

3431-111 

3431-481 

3431-852 

3432-222 

3432-593 

3432-963 

926 

927 

3433-333 

3433-704 

3434-074 

3434-444 

3434-815 

3435-185 

3435-556 

3435-92t> 

3436-296 

3436-667 

927 

928 

3437-037 

3437-407 

3437-778 

3438-148 

3438-519 

3438  889 

3439-259 

3439-630 

3440- 

3440-370 

928 

929 

3440741 

3441-111 

3441-481 

3441-852 

3442-222 

3442-593 

3442-90:3 

3443-333 

3443-704 

3444-074 

929 

930 

3444-444 

3444-816 

3445-185 

3445-556 

3445-920 

3446-296 

3446-667 

3447-037 

3447-407 

3447-778 

930 

931 

3448-148 

3448-519 

3448-889 

3449-259 

3449-630 

3450- 

3450-370 

3450-741 

3451-111 

3451-481 

931 

932 

3451-852 

3452-222 

3452593 

3452-963 

3453-333 

3453-704 

3454-074 

3454-444 

3454-815 

3455-185 

932 

933 

3455-556 

3455926 

3456296 

3456-667 

3457-037 

3457-407 

3457-778 

3458-148 

3458-519 

3458-889 

933 

934 

3459-259 

3459-630 

3460- 

3460-370 

3460-741 

3461-111 

3461-481 

3461-852 

3462-222 

3402-593 

934 

935 

3462-963 

3463-333 

3463-704 

3464-074 

3464-444 

3464-815 

3465-185 

3465-556 

3465-92*. 

3466-290 

935 

936 

3466667 

3467-037 

3467-407 

3467-778 

3468-148 

3468-519 

3468-889 

3469-259 

3469-630 

3470- 

936 

937 

3470-370 

3470-741 

3471-111 

3471-481 

3471-852 

3472-222 

3472-593 

3472-963 

3473-333 

3473-704 

937 

938 

3474-074 

3474-444 

3474-815 

3475-185 

3475-556 

3475-926 

3476-296 

3476-667 

3477-037 

3477-407 

938 

939 

3477-778 

3478-148 

3478-519 

3478-889 

3479-259 

3479-630 

3480- 

3480-370 

3480-741 

:348l-lll 

939 

940 

3481-481 

3481-852 

3482-222 

3482-593 

3482-963 

3483-333 

3483-704 

3484-074 

3484-444 

3484-815 

940 

941 

3485-185 

3485-556 

3485-926 

3486-296 

3486-667 

3487-037 

3487-407 

34S7-77S 

3488-148 

3488-519 

941 

942 

3488-889 

3489-259 

3489-630 

3490- 

3490-370 

3490-741 

3491-111 

3491-481 

3491-852 

3492-222 

942 

943 

3492-593 

3492-963 

3493-333 

3493-704 

3494-074 

3494-444 

3494-815 

3495-185 

3495-556 

3495-926 

943 

944 

3496-296 

3496  667 

3497-037 

3497-407 

3497-778 

3498-148 

3498-519 

3498-889 

3499-259 

3499-630 

944 

945 

3500- 

3500-370 

3500-741 

3501-111 

1501-481 

3501-852 

3502-222 

3502-593 

3502-5)63 

3503-333 

945 

946 

3503-704 

3504-074 

3504-444 

3504-815 

3505185 

3505-55b 

3505-926 

3506-296 

3506-667 

3507-037 

946 

947 

3507-407 

3507-77S 

3508-148 

3508-519 

3508-889 

3509-259 

3509-630 

3510- 

3510-370 

3510-741 

947 

948 

3511-111 

3511-481 

3511-852 

3512-222 

3512-593 

3512-963 

3513-333 

3513-704 

3514-074 

3514-444 

948 

949 

3514-815 

3515-185 

3515-556 

3515-926 

3516-296 

3516-667 

3517-037 

3517-407 

3517-778 

3518-148 

949 

950 

3518-519 

3518-889 

3519-259 

3519630 

3520- 

3520-370 

3520-741 

3521-111 

3521-481 

3521-852 

950 

951 

3522-222 

3522-593 

3522-963 

3523-333 

3523-704 

3524-074 

3524-444 

3524-815 

3525-185 

3525-556 

951 

952 

3525-926 

3526-296 

3526-667 

3527-037 

3527-407 

3527-778 

3528-148 

3528-519 

3528-889 

3529-259 

952 

953 

3529-630 

3530- 

3530-370 

3530-741 

3531-111 

3531-481 

3531-852 

3532-222 

3532-593 

3532-963 

953 

954 

3533-333 

3533-704 

3534-074 

3534-444 

3534-815 

3535-185 

3535-556 

3535926 

3536-296 

3536-667 

954 

955 

3537-037 

3537-407 

3537-778 

3538-148 

3538-519 

3538-889 

3539-259 

3539-630 

3540- 

3540-370 

955 

956 

3540-741 

3541-111 

3541-481 

3541-852 

3542-222 

3542-593 

3542-963 

3543-333 

3543-704 

3544-074 

956 

957 

3544-444 

3544-815 

3545-185 

3545-556 

3545-926 

3546-296 

3546-667 

3547-037 

3547-407 

3547-778 

957 

958 

3548-148 

3548-519 

3548-889 

3549-259 

3549-630 

3550- 

3550-370 

3550-741  3551-111 

3551-481 

958 

959 

3551-852 

3552-222 

3552-593 

3552-963 

3553-333 

3553-704 

3554-074 

3554-444  3554-815 

3555-185 

959 

960 

3555-556 

3555-926 

3556-296 

3556-667 

3557-037 

3557-407 

3557-778 

3558-148  3558-519 

3558-889 

9tiO 

M.A 

•O 

•1 

•3 

•3 

•* 

•5 

•6 

•7 

•  8 

•9 

M.A. 

MEAN  AREAS  9O1  to  9GO. 

RULES   FOR   THE    MEASUREMENT    OF    EARTHWORKS.  189 

CVJSIC  YARDS  TO  ME  AX  AREAS  FOR  1OO  FEET  IN  LENGTH. 


M.A. 

•0 

•  1 

•3 

•3 

•4: 

•5 

•G 

•7 

•  8 

•9 

M.A. 

9G1 

3559-259 

3559-630 

3560- 

3560-370 

3560-741 

3561-111 

3561  481 

3561-852 

3562-222 

3562-593 

961 

9f>2 

3562-963 

3563-333 

3563-704 

3564-074 

3564-444 

3564-815  3565-185 

3565-556 

3565-926 

,3566-266 

962 

963 

3566-667 

3567-037 

3567-407 

3567-778 

3568-148 

3568-519 

3568-889 

3569-259 

3569630 

3570- 

9613 

964 

3570-370 

:3570-741 

3571-111 

3571-481 

3571-852 

•3572-222 

3572-593 

3572-963 

3573-333 

3573-704 

964 

9(55 
906 

3574-074 
3577-778 

3574-444 

3578-148 

3574-815 
3578-519 

3575-185 

3578-889 

3675-556 
3579-259 

3575-926 
3579-630 

3576-296 
3580- 

3576-667 
3580-370 

3577-037 
3680-741 

3577407 
3581-111 

965 
966 

9137 

3581-481 

3581  -852 

3582-222  3582-593 

J582-963 

3583333 

3583-704 

3584-074 

3584444 

3584-815 

967 

968 
969 

3585-185  3585-556 
3588-889  35*9"259 

3585926 
3589-630 

3586-296 
3590 

3586-667 
3590-370 

3587-037 
3590-741 

3587-407 
3591-111 

3587-778 
3591-481 

".588-148 
3591-852 

3588-519 
3592-222 

968 
969 

970 

3592-593 

3592-903 

3593-333 

3593-704 

3594-074 

3594-444 

o594-815 

3595-185 

3595-556 

3595-926 

970 

971 

3596-296 

3596-667 

3597-037 

3597-407 

3597-778 

3598-1  48 

3598-519 

£598-889 

3599-259 

3599-630 

971 

972 

3600- 

3600-370 

3600-741 

3601-111 

3601-481 

3601-852 

3602-222 

3602-593 

3602963 

3603333 

972 

973 

3603-704 

3604-074 

3604-444 

3604-815 

3605-185 

3605-556 

3605926 

3606-296 

3606-667 

3607-037 

973 

974 

1607-407 

3607778 

360S-148J  3608-519 

3808*888 

3609259 

3609-630 

3610- 

3610-370 

%10-741 

974 

975 

3611-111 

3611-481 

3611-852  3612-222 

3612-593 

3612963 

3613  333 

3613-704 

3614074 

'3614-444 

975 

976 

3614-815 

3615  185 

3615-556  3615-926 

36143-296 

3616-667 

3617-037 

3617-407 

3617-778 

3618-148 

976 

977 

3618-519 

3618-8S9 

3619-259  3619-630 

3620- 

3620-370 

3620-741 

3621-111 

3621-481 

3621-852 

977 

978 

3622-222 

3622-593 

3622-963  3623  :;:;:! 

3623-704 

3024-074 

3624-444 

3624-815 

3625-185 

3625-556 

978 

979 

3625-926 

3<526"296 

3626-667 

3627-037 

3627-407 

3627-778 

3628-148 

3628-519 

3628-889 

3629-259 

979 

980 

3629-030 

3630- 

3630-370 

3630-741 

3631-111 

3631-481 

3631-852 

3632-222 

3632-593 

3632-963 

980 

981 

3633-333 

3633-704 

3634-074 

3634-444 

3634-815 

3635-185 

36:35-556 

3635-926 

3636-296 

3636-667 

9R1 

982 

3637037 

3637-407 

3637778 

3638148 

3638-519 

3638-889 

3639-259 

3039-630 

3640- 

3640370 

982 

933 

3640741 

3641-111 

3641-481 

3641-852 

3642-222 

3642-593 

3642-903 

3643-333 

3643704 

3644074 

983 

984 

3644-444 

3644-815 

3645-185 

3645-556 

3645-926 

3646-296 

3646-667 

3647-037 

3647-407 

3647-778 

984 

985 

3648-148 

3648-519 

3648-889 

3649-259 

3649-630 

3650- 

3650-370 

3650-741 

3651-111 

3651-481 

985 

9«6 

3651-852 

3652-222 

3652-593 

3652-963 

3653-333 

3663-704 

3654-074 

3654-444 

3654-815 

3655-185 

986 

987 

3655-556 

3655-926 

3656-296 

3656-667 

3657037 

3657-407 

3657-778 

3658-148 

3658-519 

3ar)8  889 

987 

988 

3659-259 

3659-630 

3600- 

3660-370 

3660741 

3661-111 

3661-481 

3661-852 

3662-222 

3662593 

988 

989 

3662-963 

3663-333 

3663-704 

3664-074 

3664-444 

3664-815 

3665-1  8f 

3665-556 

3665-926 

3666-296 

989 

9110 

3666-667 

3667-037 

3067-407 

3667-778 

3668'  148 

3668-519 

3668-889 

3669-259 

3669-630 

3670- 

990 

991 

3670-370 

3670741 

3671-111 

3671-481 

3671852 

3672-222 

3672-593 

3672-963 

3673-333 

3673-704 

991 

992 

3674-074 

3674-444 

3674-81) 

3675-185 

3675-55* 

3675-926 

3676-296 

3676-667 

3677-037 

3677-407 

992 

993 

3677-778 

3678-148  3678-519 

3678  889  3679  259 

3679-630 

3680- 

3680-370 

3680  741 

3681-111 

993 

994 

3681-481 

3681-852  1  3682-222 

3682-593  3682-963 

3683-33H 

3683-704 

5684074 

3684-444 

3684815 

994 

995 

3685-185 

3685-556 

3685-926 

3686-296,3686-667 

3687-037 

3687407 

3687*778 

3688-148 

3688-519 

995 

996 

3688-889 

3689-259 

3889-630 

3690-       13690-370 

3690-741 

3691-111 

3691-481 

3691-852 

3692-222 

996 

907 
998 

3692-593 
3696-296 

3692-963  3693-333 
3696-667  3697-037 

3693704  3'  94  074 
3697(407  3697-77^ 

3694-444 

3698-148 

3694-81  5  13695-1  85 
3698-519  3698-889 

3695-556 
3699-259 

3695926 
3699630 

997 
998 

999 

3700- 

3700-370 

3700-741 

3701-111 

3701-481 

3701-852 

3702-222 

3702-593 

3702-963 

3703-333 

999 

1000 

3703-704 

3704-074 

3704-444 

3704-815 

3705-185 

3705-556 

3705-920 

3706-296 

3706-667 

3707-037 

1000 

M.A 

•0 

•1 

•3 

•3 

•4 

•5 

•6 

•7 

•8 

•9 

M.A. 

MEAN  AREAS   961  to  1OOO. 

NOTE.  —  This  Table  having  been  carefully  computed  by  the  Author,  through  the 
usual  method  of  successive  additions,  and  verified  in  the  manuscript,  was  set  up  by  a 
skilful  printer,  and  the  proofs  examined,  and  re-examined,  until  they  were  thought  to 
be  free  from  error;  finally,  the  plates  were  cast,  and  the  revises  taken  from  them  sub- 
mitted, page  by  page,  to  the  scrutiny  of  a  competent  Civil  Engineer,  who  examined  the 
whole,  figure  by  figure,  and  ultimately  reported  but  few  slight  mistakes,  which  were 
immediately  corrected  in  the  plates  themselves;  so  that  every  precaution  having  been 
taken  to  secure  accuracy: — the  Author  feels  justified  in  declaring  his  belief,  that  the 
Table  above  t«  entirely  clear  of  any  material  error. 


SCIENTIFIC  BOOKS, 

PUBLISHED    BY 


23  MURRAY  STREET,  and  27  WARREN  STREET, 
IE  -W    ^r  o  : 


PLATTNER'S  MANUAL  OF  QUALITATIVE  AND  QUANTI- 
TATIVE ANALYSIS  WITH  THE  BLOWPIPE.  From  the  last 
German  edition,  revised  and  enlarged.  By  Professor  TH.  KICHTEB. 
Translated,  by  Prof.  HENRY  B.  CORNWALL,  E.  M.  Illustrated  with 
eighty-seven  woodcuts  and  one  lithographic  plate.  560  pages,  Svo. 
Cloth.  $7.50. 

LOWELL  HYDRAULIC  EXPERIMENTS-being  a  selection  from 
Experiments  on  Hydraulic  Motors,  on  the  Flow  of  Water  over  Weirs, 
and  in  Open  Canals  of  Uniform  Rectangular  Section,  made  at  Lowell, 
Mass.  By  J.  B.  FRANCIS,  Civil  Engineer.  Third  edition,  revised 
and  enlarged,  including  many  New  Experiments  on  Gauging  Water 
in  Open  Canals,  and  on  the  Flow  through  Submerged  Orifices  and  Di- 
verging Tubes.  With  23  copperplates,  beautifully  engraved,  and  about 
100  new  pages  of  text.  1  vol.,  4to.  Cloth.  $15. 

FRANCIS  (J.  B.)  ON  THE  STRENGTH  OF  CAST-IRON  PIL- 
LARS, with  Tables  for  the  use  of  Engineers,  Architects  and  Build- 
ers. Svo.  Cloth.  $2. 

USEFUL  INFORMATION  FOR  RAILWAY  MEN.  Compiled  by 
W.  G.  HAMILTON,  Engineer.  Fourth  edition,  revised  and  enlarged. 
570  pages.  Pocket  form.  Morocco,  gilt.  $2. 

WEISBACH'S  MECHANICS.  New  and  revised  edition.  A  Manual 
of  the  Mechanics  of  Engineering,  and  of  the  Construction  of  Ma- 
chines. By  JULIUS  WEISBACH,  PH.  D.  Translated  from  the  fourth 
augmented  and  improved  German  edition,  by  ECKLEY  B.  COXE,  A.  M., 
Mining  Engineer.  Vol.  I. — Theoretical  Mechanics.  1  vol.  Svo, 
1100  pages,  and  902  woodcut  illustrations,  printed  from  electrotype 
copies  of  those  of  the  best  German  edition.  $10. 

ABSTRACT  OF  CONTENTS. — Introduction  to  the  Calculus — The  General  Principles  of 
Mechanics — Phoronomics,  or  the  Purely  Mathematical  Theory  of  Motion — Mechanics, 
or  the  General  Physical  Theory  of  Motion — Statics  of  Rigid  Bodies — The  Application 
of  Statics  to  Elasticity  and  Strength — Dynamics  of  Rigid  Bodies — Statics  of  Fluids — 
Dynamics  of  Fluids — The  Theorj7  of  Oscillation,  etc. 

"  The  present  edition  is  an  entirely  new  work,  greatly  extended  and  very  much  improved.  It 
forms  a  text-book  which  must  find  its  way  into  the  hands  not  only  of  every  student,  but  of  every 
engineer  who  desires  to  refresh  his  memory  or  acquire  cle:tr  ideas  on  doubtful  points." — The  Techno- 
logist. 

1 


D.  VAN  NOSTRAND'S  LIST  OF  SCIENTIFIC   BOOKS. 

A  TREATISE  ON  THE  PRINCIPLES  AND  PRACTICE  OP 
LEVELLING.  Showing  its  application  to  purposes  of  Railway  Engi- 
neering and  the  Construction  of  Roads,  etc.  By  FREDERICK  W. 
SIMMS,  C.  E.  From  the  fifth  London  edition,  revised  and  corrected, 
with  the  addition  of  Mr.  Law's  Practical  examples  for  Setting  Out 
Railway  Curves.  Illustrated  with  three  lithographic  plates  and  nu- 
merous woodcuts.  8vo.  Cloth.  $2.50. 

PRACTICAL  TREATISE  ON  LIMES,  HYDRAULIC  CEMENTS, 
AND  MORTARS.  Containing  reports  of  numerous  experiments 
conducted  in  New  York  City,  during  the  years  1858  to  1861,  inclusive. 
By  Q.  A.  GILLMORE,  Brig. -General  U.  S.  Volunteers,  and  Major  U. 
S.  Corps  of  Engineers.  With  numerous  illustrations.  Third  edition. 
8vo.  Cloth.  $4. 

COTGNET  BETON,  and  other  Artificial  Stone.  By  Q.  A.  GILLMORE, 
Major  U.  S.  Corps  Engineers.  Nine  plates  and  views.  8vo.  Cloth. 

$2.50. 

A  TREATISE  ON  ROLL  TURNING  for  the  Manufacture  of  Iron. 
By  PETER  TUNNER.  Translated  and  adapted.  By  JOHN  B.  PEASE, 
of  the  Pennsylvania  Steel  Works.  With  numerous  engravings  and 
woodcuts.  1  vol.  8vo,  with  1  vol.  folio  of  plates.  Cloth.  $10. 

MODERN  PRACTICE  OF  THE  ELECTRIC  TELEGRAPH.  A 
Hand  Book  for  Electricians  and  Operators.  B}'  FRANK  L.  POPE. 
Fifth  edition.  Revised  and  enlarged,  and  fully  illustrated.  8vo. 
Cloth.  $2. 

TREATISE  ON  OPTICS  ;  or,  Light  and  Sight  Theoretically  and 
Practically  Treated,  with  the  application  to  Fine  Art  and  Industrial 
Pursuits.  By  E.  NUGENT.  With  103  illustrations.  12mo.  Cloth.  $2. 

THE  BLOW-PIPE.  A  System  of  Instruction  in  its  practical  use, 
being  a  graduated  course  of  Analysis  for  the  use  of  students,  and  all 
those  engaged  in  the  Examination  of  Metallic  Combinations.  Second 
edition,  with  an  appendix  and  a  copious  index.  By  Professor  GEORGE 
W.  PLYMPTON,  of  the  Polytechnic  Institute,  Brooklyn.  12mo. 
Cloth.  $2. 

KEY  TO  THE  SOLAR  COMPASS  and  Surveyor's  Companion; 
comprising  all  the  Rules  necessary  for  use  in  the  Field.  By  W.  A. 
BURT,  U.  S.  Deputy  Surveyor.  Second  edition.  Pocket  Book  form. 
Tucks.  $2.50. 

MECHANIC'S  TOOL  BOOK,  with  practical  rules  and  suggestions, 
for  the  use  of  Machinists,  Iron  Workers,  and  others.  By  W.  B. 
HARRISON,  associated  editor  of  the  "American  Artisan."  Illus- 
trated with  44  engravings.  12rno.  Cloth.  $2.50. 

MECHANICAL  DRAWING.  A  Text-Book  of  Geometrical  Drawing 
for  the  use  of  Mechanics  and  Schools,  in  which  the  Definitions  and 
Rules  of  Geometry  are  familiarly  explained  ;  the  Practical  Problems 
are  arranged,  from  the  most  simple  to  the  more  complex,  and  in  their 
description  technicalities  are  avoided  as  much  as  possible.  With  illus- 
trations for  Drawing  Plans,  Sections,  and  Elevations  of  Buildings  and 
Machinery ;  an  Introduction  to  Isometrical  Drawing,  and  an  Essay 
on  Linear  Perspective  and  Shadows.  Illustrated  with  over  200  dia- 
grams engraved  on  steel.  By  WM.  MINIFIE,  Architect.  Eighth 
edition.  With  an  appendix  on  the  Theory  and  Application  of  Colors. 
I  vol.,  8vo.  Cloth.  $4. 

2 


D.  VAN  NOSTRAND'S  LIST  OF  SCIENTIFIC  BOOKS. 

A  TEXT  BOOK  OF  GEOMETRICAL  DRAWING.  Abridged  from 
the  octavo  edition,  for  the  use  of  Schools.  By  "W.  MIMFIE.  Illustra- 
ted with  48  steel  plates.  New  edition,  enlarged.  1  vol..  12mo. 
Cloth.  $2. 

"It  is  well   adapted   as  a  text-book  of  drawing  to   be  used  in  our  High  Schools  and  Academies 
where  this  useful  branch  of  the  fine  arts  has  been  hitherto  too  much  neglected." — Boston  Journal. 

TREATISE  ON  THE  METALLURGY  OF  IRON.  Containing 
outlines  of  the  History  of  Iron  manufacture,  methods  of  Assay,  and 
analysis  of  Iron  Ore,  processes  of  manufacture  of  Iron  and  Steel,  etc., 
etc.  By  H.  BAUERMAN.  First  American  edition.  Revised  and  en- 
larged, with  an  appendix  on  the  Martin  Process  for  making  Steel,  from 
the  report  of  Abram  S.  Hewitt.  Illustrated  with  numerous  wood 
engravings.  12mo.  Cloth.  $2.50. 

IRON  TRUSS  BRIDGES  FOR  RAILROADS.  The  Method  of 
Calculating  Strains  in  Trusses,  with  a  careful  comparison  of  the  most 
prominent  Trusses,  in  reference  to  economy  in  combination,  etc.,  etc. 
By  Brevet  Colonel  WILLIAM  E.  MERRILL,  U.  S.  A.,  Major  Corps 
of  Engineers.  With  illustrations.  4to.  Cloth.  $5. 

THE  KANSAS  CITY  BRIDGE.  With  an  account  of  the  Regimen 
of  the  Missouri  River,  and  a  description  of  the  Methods  used  for 
Founding  in  that  River.  By  0.  CHANUTE,  Chief  Engineer,  and 
GEORGE  MORISON,  Assistant  Engineer.  Illustrated  with  five  litho- 
graphic views  and  12  plates  of  plans.  4to.  Cloth.  $6. 

CLARICE  (T.  C.)  Description  of  the  Iron  Railway  Bridge  across  the 
Mississippi  River  at  Quincy,  Illinois.  By  THOMAS  CURTIS  CLARKE, 
Chief  Engineer.  Illustrated  with  numerous  lithographed  plans. 
1  vol.,  4to7  Cloth.  $7.50. 

AUCHINCLOSS.  Application  of  the  Slide  Valve  and  Link  Motion  to 
Stationary,  Portable,  Locomotive,  and  Marine  Engines,  with  new  and 
simple  methods  for  proportioning  the  parts.  By  WILLIAM  S.  AUCHIN- 
CLOSS, Civil  and  Mechanical  Engineer.  Designed  as  a  handbook  for 
Mechanical  Engineers,  Master  Mechanics,  Draughtsmen,  and  Stu- 
dents of  Steam  Engineering.  All  dimensions  of  the  valve  are  found 
with  the  greatest  ease  by  means  of  a  PRINTED  SCALE,  and  propor- 
tions of  the  link  determined  without  the  assistance  of  a  model.  Illus- 
trated'by  37  woodcuts  and  21  lithographic  plates,  together  with  a  cop- 
perplate engraving  of  the  Travel  Scale.  1  vol.,  8vo.  Cloth.  $£. 

GLYNN  ON  THE  POWER  OF  WATER,  as  applied  to  drive  Flour 
Mills  and  to  give  motion  to  Turbines  and  other  Hydrostatic  Engines. 
By  JOSEPH  GLYNN,  F.  R.  S.  Third  edition,  revised  and  enlarged, 
with  numerous  illustrations.  12mo.  Cloth.  $1.25. 

HUMBER'S  STRAINS  IN  GIRDERS.  A  Handy  Book  for  the  Cal- 
culation of  Strains  in  Girders  and  Similar  Structures,  and  their 
Strength,  consisting  of  Formulae  and  Corresponding  Diagrams,  with 
numerous  details  for  practical  application.  By  WILLIAM  HUMBER. 
1  vol.,  18  mo.  Fully  illustrated.  Cloth.  $2.50. 

***  Any  of  the  above  Books  sent  free,  by  mail,  on  receipt  of 
price. 

JS^r*  My  new  Catalogue  of  American  and  Foreign  Scientific 
Books,  72  pages,  8vo.,  sent  to  any  address  on  the  receipt  of  Ten 
cents. 

3 


18SO. 


WM.  J.  YOUNG  &  SONS, 

SUCCESSORS    TO 

Win.  J.  Yonng.  WE  J,  Young  &  Son.  WE  J.  Yonng  &  Co, 


I 


BANDFACTDRERS, 


TRANSITS, 

tEVEia. 

SGtAR  COMPASSES,  ETC., 

Of  most  approved  construction,  with  latest  improvements  in 
both  Instruments  and  Telescopes. 

Instruments  constructed  on  scientific  principles,  guaranteed 
to  afford  greatest  strength  and  durability  with  least  weight. 

Tapes,  Chains,  Draughting  Instruments,  and  all  supplies  for 
Field  and  Office. 

Especial  attention  of  Engineers  called  to  Profile  Field 
Boole. 


T.  R.  CALLENDER. 


JOHN  P.  HUNT 


T  R.  CALLENDER  &'  CO, 

Publishers,  Booksellers  and  Newsdealers, 

N.  W.  COR.  THIRD  AND  WALNUT  STREETS, 
PHILADELPHIA. 

GENERAL  SUBSCRIPTION  AGENCY 

For  American,  English  and  French 

Newspapers  and  Magazines. 


Subscriptions  received  for  all  the  American  and  Foreign  Scientific 
Magazines  and  Periodicals.  Among  them  the  following,  at  the  an- 
nexed prices,  United  States  currency  : — 

Appleton's  Monthly $5  00 

Chemical  News 12  00 

Colburn's  United  Service  Maga- 
zine  22  00 

English  Mechanic 6  00 

English  Mechanic  and  Mirror  of 
Science 6  00 

Geological  Magazine 10  00 

London  Artisan 7  00 

London  Builder 13  00 

London  Engineer,  including  Sup- 
plements   12  00 

London  Engineering 12  00 

London,  Edinburgh  and  Dublin 
Journal  of  Phil.  Science 21  00 

London  Mining  Magazine 14  00 

London  Mechanics'  Magazine 12  00 

London    Monthly   Microscopical 
Journal . ..19  00 

London  Nature 6  00 

London  Nautical  Magazine 7  00 

London  Popular  Science  Review.  5  00 

London  Mining  Journal 14  00 

London  Railway  Times 14  00 

London   Chemist  and  Druggist, 
Monthly 5  00 

London  Geological  Magazine 9  00 

Micriscopal  Monthly 9  00 

Quarterly  Journal  of  Science 10  00 

"              "            Geological 
Society 10  00 

Quarterly  Journal  of  Micriscopical 
Science 10  00 

Scientific  American 3  00 

Van  Nostrand's  Ecletic  Engineer- 
ing Magazine 5  00 

Workshop  (The) 5  50 


Album  indusiriel  et  de  la  chaus- 

sure $  5  00 

Ameublement  (!')  et  Putile  reunis. 

Noir 7  50 

do.    Couleur 12  50 

Annales  du  Conservato)re  imperial 

des  arts  et  metiers 10  00 

Architecture  allemande  avi   XIX 

me 12  00 

Batiment  (le.) — Journal  general 
des  travaux  publics  et  prives  et 
leMoituer  des  batiments  reunis.  7  50 

Garde-Meuble  (le.)    Noir 5  65 

Couleur 9  00 

Gazette  des  architectes  et  du  bati- 

ment 12  50 

Guide  [le]  du  carrossier 12  00 

Journal  de  Menuiserie 12  00 

Journal  des  Brasseurs 7  00 

Journal  des  Chapeliers  et  de  la 

Chapellerie.. 7  00 

Journal-Manuel  de  Peintures  ap- 
pliquees  a  la  decoration  des 
monuments  apartments,  maga- 

sins,  etc.,  etc., 11  00 

Magisin  de  meubles,  publiant,  48 

planches  par  an.     Moir 7  50 

Couleur 15  00 

Moniteur  de  la  bijouterie 7  00 

Moniteur  de  la  brasserie 6  00 

Moniteur  de  la  chapellerie  ou 
1'Echo  des  applications  qui  se 

rapportent  a  cette  Industrie 6  00 

Moniteur  de  1'horlogerie 7  00 

Moniteur  des  Architectes 12  50 

Moniteur  des  marbriers-sculpteurs  7  00 

Revue  general  de  1'architecture  et 

.   des  travaux  publics 20  00 


Catalogues  of  Scientific  and  Mechanical  Works  furnished  on  appli- 
cation. 

T.  R.  CALLENDER  &  CO. 


UNIVERSITY  OP  CALIFORNIA  LIBRARY 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


Feb'52W 


30m-l,'15 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


